Post-Modern Algebra

l in T. Proof. Let A be the set of subsets of G of order p'. Note that A is a G-subset of the G-set 2 G of Exercise 1H. Let T be a set of representatives for the orbits A/G. By the Structure Theorem for G-sets, {A, G) = LSBT(GS\G,G). For S in T, one has GSSJ = SJ, so that SJ is a G| p subset of the right G^-set (G, G£p) of (1.3.3). As such, SJ is a disjoint union of orbits, namely right cosets of Gs. Thus \GS\ divides | 5 y | = p'. In particular, | G S \ G | = m » Gs <= A <=» (GS\G,G) <, (A, G).

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83

Indeed, H -» (Hy/G, G) yields a bijection from Sbp<(G) to the set of irreducible/«-element G-subsets of (/I, G). Thus \A\ = HSeT\Gsy/G\(pm) LHesb «(0)1^X^1 = ISb «(G)| -m. Suppose m = pfl with p not dividing /. Then '\A\ = p'^l ■ (p'+H - 1 ) . . . (p'+fl - ip' - l))/p'(pe - 1 ) . . . 1 = mTlfcKp'+'l - k)/k(pm)m, since KpmfcKp'+'l - k)/k. Thus ISbp.tG)! •m(pm)m, whence the result. D Corollary 3.3.2. Let G be afinitegroup. (a) (Cauchy's Theorem.) If a prime p divides \G\, then G contains an element of order p. (b) For a prime p, the group G is a p-group if and only if \G\ is apowerofp. (c) For a prime p, if \G\ = p'm with p not dividing m, then Sylp(G) = Sb,.(G). Proof, (a) Take a non-identity element of an element of Sbp(G). (b) If \G\ is not a power of p, so some other prime q divides \G\, then G contains an element of order q, by (a). Conversely, if G contains an element x whose order r is not a power of p, then Lagrange's Theorem [Exercise 1.3J(d)] shows that |<{JC}>| = r divides \G\, which cannot then be a power of p. (c) The proof follows from (b). D Theorem 3.3.3 (Sylov's Theorem). Let G be a finite group, and let p be a prime. (a) In Z, one has |Sylp(G)|>l. In particular, Sylp(G) is non-empty. (b) The inner automorphism group of G acts transitively on Syl (G). In particular, |Sylp(G)| divides \G\. Proof, (a) By Corollary 3.3.2 (c) and Proposition 3.3.1. (b) Let \G\ = p'm with p not dividing m. Let P and Q be Sylov p-subgroups of G. Let K be the subgroup of Mit G generated by L(P) and R(Q). Consider the action of K on G. For g in G, the size of the orbit of g is \PgQ\ = \g-'PgQ\ = \PgQ\ = \Pg\ ■ \Q\/\P* n Q\ [cf. Exercise 1.3L(c)]. If P* were never to agree with Q, then the size of all the orbits, and hence their sum |G|, would be divisible by p'+1. D Let (X,G) be a permutation group with permutation character IT. A non-trivial subgroup H of G is regular if v(h) = 0 for all non-identity elements of H. If TV is a regular normal subgroup of G (e.g. the Vierergruppe in S4: Exercise 3.2E) and e is an element of X, then Af - {1} becomes a Ge-subset of the Ge-set (G,Ge) whose representation is the restriction T :Ge-* Inn G of (2.4.3). One then obtains a Ge-isomorphism (3.3.1)

(N-{l},Ge)

- (X-

{e},Ge);n * en.

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CHAPTER I GROUPS AND QUASIGROUPS

If (X, G) is fc-transitive, for k > 1, then Ge is a subgroup of the automorphism group of N that is (k — D-transitive on N - {1} (Exercise 3K). This forces TV to be a p-group for some prime p: Proposition 3.3.4. Let A be a subgroup of the automorphism group of a non-trivialfinitegroup N. (a) If (N - {1}, A) is l-transitive, then N is a direct power of (Zp, + ) for some prime p. (b) / / (N - {1}, A) is 2-transitive, then p-2orNis isomorphic to (Z3, + ). (c) If (N — {1}, A) is 3-transitive, then N is isomorphic to (Z3, +) or the Vierergruppe V. (d) It is impossible for (N - {1}, A) to be 4-transitive. Proof, (a) Since N is non-trivial, some prime p divides its order. By Cauchy's Theorem (Corollary 3.3.2(a)), N has an element of order p. Since orders are preserved by automorphisms, each non-identity element of N has order p. By Corollary 3.3.2(b), the order of N is some power p' of p. By Corollary 2.5.5, the center Z(N) of N is non-trivial. Since Z(N) - {1} is invariant under the transitive group A of automorphisms (Exercise 2.5G), it follows that N = Z(N) is abelian. Let H be a subgroup of N with x e AT H. Then H and <JC> are normal subgroups of (H, x) = H{x) with H n <*> = {1}. By Section O 4.4, the permutable (Proposition 2.2.3) congruences on {H,x) determined by the normal subgroups H and <*> decompose (H,x) ssHx <JC>. Since <*> = (Zp, +), it follows by induction that N s (Zp, +) r . (b) Since Af is abelian, the inversion map J of (1.3.2) is an automorphism of Af. Moreover, the subsets and A of Aut Af are permutable (cf. Exercise 2.1G). Consider the <7>-set (N - {1}, < / » . The kernel of the projection (Af - {1}) -* (N - {l})/ is an ^4-subset of (AT - {l})2. Since it is reflexive, it is either the diagonal, in which case the -orbits are trivial and p = 2, or else it is (Af - {l})2, in which case there is only one -orbit on N — {!), necessarily of order 2, and N is isomorphic to (Z3, +). (c) If p = 2, then N contains an (abstract) Vierergruppe V = {1, x, v, xy] as a subgroup. The stabilizer Ax acts 2-transitively on N — (x) (cf. Exercise 3K). Let a be the kernel of the projection N -* (x)\N. Then a n (N — (x))2 is a reflexive -¿¿-subset of (AT - (A:» 2 containing (v, xy). It thus coincides with (N - (x))2, in which case N — (x) = {v, xy) and N = V. (d) Neither (Z3, Aut(Z3,+ )) nor (V, AutF) is 4-transitive. D Corollary 33.5. Let N be a regular normal subgroup of a permutation group G of a set X of order n. (a) (b) (c) (d)

If If If It

(X,G) is 2-transitive, then n is a prime power pr and N = (Z p , +) r . (X, G) is 3-transitive, then n = 2r or n = 3. (X, G) is 4-transitive, then n = 4 or n = 3. is impossible for (X,G) to be 5-transitive.

SYMMETRY

85

Proof. If e is an element of thefc-transitiveG-set X, then (X - {e}, Ge) is (A: - l)-transitive (cf. Exercise 3K). The isomorphism (3.3.1) then yields a (it - D-transitive action of Gt on TV - {1}. The disjoint union with the singleton Ge-set {1} then yields Ge as a subgroup of the automorphism group of TV. a Theorem 3.3.6. For n > 4, the group An is simple. Proof. Let TV be a non-trivial normal subgroup of the (n - 2)-transitive permutation group An on Z„. Now An acts on Z„/TV by g :xN <-* xNg = xgg~lNg = xgN. Thus the projection Z„ -» Z„/TV is an y4„-homomorphism whose kernel is a non-trivial ^„-congruence. By Exercises 31 and 3J, the action An is simple. Thus the kernel of the projection is all of Z 2 , whence (Z„, TV) is transitive and n divides |TV|. If n = 5, then \An\ = 5!/2 = 2 2 • 3 • 5. By Proposition 3.2.2(b), there are 5!/5 = 24 permutations of type 10203°4°51 in S5, and hence in A5. Each Sylov 5-subgroup of As contains 4 such permutations (along with the identity), so there are 6 Sylov 5-subgroups of A¡. By Sylov's Theorem 3.3.3(b), they are all conjugate. Now TV has order a multiple of 5, so by Cauchy's Theorem it contains at least one of these Sylov subgroups. But since TV is normal, it contains all 6, i.e. |Syl5(/V)| = 6. By Sylov's Theorem 3.3.3(b), it follows that 6 also divides |TV|, whence |TV| is 2 • 3 • 5 or 2 2 • 3 • 5. By Cauchy's Theorem again, TV contains at least one Sylov 3-subgroup of As. There are 5!/3 • 2 = 20 permutations of type 1°203,405° in S5 and As, pairing off with the identity to make 10 Sylow 3-subgroups. All are contained in the normal subgroup TV, so TV contains 20 cycles of length 3 and 24 of length 5. In particular, 30 < |TV|, so |TV| = 60 and TV is improper. In other words, A5 is simple. Now for n > 5, assume by induction that An_x is simple. Let TV be minimal amongst the non-trivial normal subgroups of An. Recall that An_l is the stabilizer of n in An. If TV n An_x were trivial, the stabilizer TV„ of n in TV would be trivial. By Proposition 2.4.5, all the stabilizers TV, in the transitive action (Z„, TV) would then be trivial, whence TV would become a regular normal subgroup of the (n - 2)-transitive permutation group An. By Corollary 3.3.5(d), this would force n = 6. But since 6 is not a prime power, the demand of Corollary 3.3.5(a) would be violated. Thus TVn^l,,., is a non-trivial normal subgroup of An_1. By the induction assumption, N C\An_x =An_l, so that TV„ =An_i. Since (Z„, TV) is transitive, Proposition 1.3.2 yields |TV| = |Z„| • |TV„| = n ■ (n - 1)! = n\ = \An\, so that TV = A„ and A„ is simple. D EXERCISES 3.3A. Let TV be a normal subgroup of a group G. For a prime number p, show that G is a /»-group if and only if both TV and G/N are /»-groups. 3.3B. Show that the product Z -> n"_,Z < r > of the maps n >-> n<2'> injects.

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CHAPTER I GROUPS AND QUASIGROUPS

33C. By considering the factor 6 of \A4\, show that Cauchy's Theorem does not extend to composite divisors of the order of a finite group. 3.3D. By considering (2.2), show that Proposition 3.3.1 does not extend to quasigroups. 33E. Determine the number of Sylov 2-subgroups of 5 4 . Show that one of them is equal to Z>4. 33F. Show that the action of a Sylov p-subgroup of Sp2, for prime p, is similar to the wreath product action Cp\ Cp. 33G. Let the order of a group G be the product pq of primes p and q, with p > q. Show that G has a normal subgroup of order p. 33H. For each odd prime p, determine the Sylov p-subgroups of the dihedral group D„. 331. Show that there is no simple group of order 56. 33J. (a) Determine the orders of the (group) conjugacy classes of the group A,. (b) Show that no proper non-singleton Inn ^45-subset of A5 containing 1 has order dividing 60. (c) Deduce that As is simple (independently of the proof of Theorem 3.3.6). 4. LOOPS, NETS, AND ISOTOPY In Example 2.4, the quasigroup Q with multiplication table (2.2) was obtained from the Latin square in (2.1) by an arbitrary assignment of the elements of Q as labels for the rows and columns of the table. Rearranging the row and column labels would yield the multiplication table of a different quasigroup, not necessarily isomorphic to the original. For example, the quasigroup Q with table (2.2) has three idempotents, namely A, B, and C. If the column headings are changed by application of the cycle (ABCV), then the new quasigroup Q' [as in (4.2) below] only has a single idempotent, namely V. Isomorphism of quasigroups is too strong a relationship to describe the intrinsic structure of a Latin square. The appropriate relationship is known as "isotopy." Recall that a quasigroup homomorphism / : (Q, ■, / , \ ) -» (/>, •, / , \ ) is a set map f:Q-*P with xf-yf = (xy)f for all x, y in Q (Exercise 2.2E). A quasigroup homotopy (/„ f2, / 3 ) : (ß, •, / , \ ) -* (P, •, / , \ ) is a triple of set maps f¡:Q-*P with (4.1)

xft -yf2 = (xy)f3

for all x, y in Q. Thus a homomorphism is a homotopy in which the three components agree. A quasigroup isotopy is a homotopy in which each compo-

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87

nent bijects. Two quasigroups ß,, Q2 are said to be isotopic, written Qx ~ Q2, if there is an isotopy ß j -> Q2. One also says that Q2 is an isotope of Qv Considering the quasigroups ß and Q' on the set {A,B,C,V} of Example 2.4, with multiplication tables Q

A B C V

A A C V B

B V B A C

Q'

V

C B A C V

c

A B C V

V B A

A C V B A

B A C V B

C V B A C

V B A C V

one sees that ß and Q' are isotopic via the isotopy (1, (ABCV), 1) : ß -» ß ' . The second component of the isotopy records the permutation of the column labels in the original table of Q. Similarly, the first component of an isotopy records symmetries of the row labels, while the third component records symmetries of the set of table entries. The arithmetic mean quasigroup (IR, ° ) of Example 2.3 is isotopic to the abelian group (R, +) via the isotopy (1,1,2):(R, °) -» (R, +) whose third component doubles its argument. An isotopy with equal domain and codomain is said to be principal if its third component is the identity map of its domain. Note that any isotopy (/i,/ 2 ,/ 3 ) factorizes as a product (fjf\ / 2 /3 - 1 .lX/ 3 , / 3 , / 3 ) of a principal isotopy with an isomorphism, or vice versa as a product (/3> fi> fi^filfi' fi1f2< D °f a n isomorphism with a principal isotopy. Now a loop (Q, •, / , \ , 1) is a quasigroup (ß, •, / , \ ) with an identity element 1 of ß satisfying (4.3)

\x=x

= xl

for all x in Q. Let (ß, •, / , \ ) be a quasigroup with element e. Define a new quasigroup structure (ß, *, / / , \ \ ) on ß by

(4.4)

(x*y = \x//y = \x\\y =

(x/(e\e))-(e\y), (x/(e\y))(e\e), (e/e)-((x/e)\y).

Then (ß, *, / / , \ \ , e) is a loop. Moreover, there is a principal isotopy (Ä(e\e),L(e),l):(ß,-,/,\)-(ß,*,//A\). Thus: Proposition 4.1. Let Q be a quasigroup with element e. Then Q is principally isotopic to a loop with identity element e. D

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CHAPTER I GROUPS AND QUASIGROUPS

One often wishes to study properties of a quasigroup that are shared by its isotopes. To this end, a relational structure is associated with a quasigroup in such a way that isotopic quasigroups correspond to isomorphic structures. Definition 4.2. Let k be an integer bigger than 2. Then a k-net is a relational structure (N,
Proposition 4.3. Let (N,(ak\l <, i <, k)) be a k-net. (a) The set {a,|l £ i <. k) is permutable. (b) For 1 ) bijects. (c) The sets N"' are all isomorphic for 1 ^ i' <, k. (d) / / \N\ > 1, tfien i * j =» a, # a¡. (e) // 1 < |ATI < oo, then (k - l) 2 <; |N|. /Voo/ (a) By the definition, a¡ ° aj = N2 = a¡° a¡. (b) Cf. Section O 4.4. (c) If N is empty, so is each N"'. Otherwise, fix e in N. Consider \{h, i, j}\ = 3 with \ <.h, i, j <, k. Then there is a bijection (4.5)

T(e,h,i,j):Na'^Na<

given by na> -* \J{t"'\{t} = na- n e"*}. (d) Suppose i =* j and a, = a¡. Then N = a¡ n a ; = a, » a ; = AT2, whence \N\ z 1. (e) Firstly, note IN"'I > 1 for each l g i £ k: otherwise, (c) and (b) would imply \N\ ^ 1. Pick e and f in N with e"1 * / " ' . For 1 < 11 <, k, pick t¡ in e"1 n / " 1 . Then t2 tk are distinct, for t¿ = f;- would imply (e,t¡) = (e, tj) e a, n a, = Ñ, whence / " ' = f/"' = e"'. Now f2"2,.... ffc"2 are distinct, for r¡"2 = f/12 would imply (r¿, ry) 6 o , n o 2 = JV. Thus |N" 2 | > k - 1 and |N| = |N"'| • IN"21 = |N"212 ;> U - l) 2 , as required. D In view of Proposition 4.3(b) and Section O 4.4, one may loosely describe a &-net as a set that is the product of any pair of a fc-element multiset of quotients. A quasigroup Q determines a 3-net Q2 = Net(ß). The underlying set of the net is the Cartesian square N = Q2, equipped with the projections 77-, : Q2 -» ß ; (*„ x2) •-» x¡. Then a, = ker 7r, for i = 1,2, while a 3 = ker ( ß 2 -» Q; (xu x2) ~* . x ^ ) . N o t e » for example, that (xlt x2) at (xu xt \ yrfi) a3 ( ^ I . ^ X using the notational convention introduced following Proposition 2.2.2, so that a , » a 3 = N 2 . Also (xv x2) al n a 3 (jCj, y2) => x2 = = AÍJ \jc,jf2 = x\ \x\y2 ^ 2 ' s o t n a t a i n a 3 = N. Similarly a2 n a 3 = 2 N and a 2 ° a 3 = N . Especially for a finite quasigroup Q, the 3-net Net(Q)

LOOPS, NETS, AND ISOTOPY

89

o—o—o—o ö—o—o—o O—O—O—Ó O—O—O—Ó («>

(»)

Figure 12. (a) The Latin square (1.2.2). (b) The corresponding net.

may be viewed as an unlabeled version of the Latin square obtained from the multiplication table by removing the borders. The a r classes correspond to the rows, so the elements of an a,-class may be joined by chains of horizontal Unes. Similarly, the elements of an a2-class, representing a column in the table, may be joined by chains of vertical lines. Finally, the elements of an a3-class, representing table locations with the same entry, may be joined by chains of diagonal lines. Figure 1.2(a) shows the Latin square determined by the multiplication table of the quasigroup Q of (2.2) or (4.2), while Figure 1.2(6) displays the corresponding 3-net Net (Q) using the conventions described. Because of these conventions, the a,-classes of a 3-net are often called the horizontal, vertical, and diagonal lines of the net, or sometimes just the i-lines, for / = 1,2,3, respectively. Two 3-nets (N,(av a2, a3)) and (N',(a[, a'2, a 3 » are isomorphic if there is a bijection f:N-»N' such that («i, n2) e a¡ => (nlf, n2f) e a¡ for each i. Thus an isomorphism of 3-nets carries horizontals to horizontals, verticals to verticals, and diagonals to diagonals. Now gjven a 3-net (N,
•x2 = *i ♦» 0 #

n P,- 1 {*,)•

The following result shows how isotopic quasigroups correspond to isomorphic 3-nets.

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Theorem 4.5. (a) Let (ß, • ) and (P, °) be isotopic quasigroups. Then the 3-nets Net(ß, • ) and Net (P, ° ) are isomorphic. (b) Let p¡:N -» JÑ and p't:N -* JÑ (for 1 :£ i £ 3) be two triples of maps from a 3-net N with a¡ = ker p¡ = ker p'^JTien the quasigroups N(pu p2, p3) = WÑ, • ) and N(p\, p'2, p'3) - (vW, °) are isotopic. Proof, (a) Suppose there is a isotopy (fx,f2,f3):(Q, • ) -» (P, °), so that xïfi°x2f2 = (xix2)f3 for xu x2 in Q. Then / : ß 2 - » P 2 ; (xl,x2)>-* (xlf1,x2f2) bijects. It will be shown that / is an isomorphism from Net(ö» ■ ) to Net(P, »). By construction, / carries horizontals to horizontals and verticals to verticals. Suppose (xux2) a3(yvy2) in Q2. Then xlf1°x2f2 = (xxx2)f3 = (yxy2)f3 = yxfi ° y2f2, as required. (b) For 1 £ ¿ *£ 3, the First Isomorphism Theorem O 3.3.1 shows that p¡: N -» ■/Ñ factorizes through an isomorphism^: N"- -» ■/Ñ and that p\ factorizes through an isomorphism p[:JVa' -» vW. Define bijections f¡ = ~p~ p¡ : \/Ñ -* JÑ. Then for x¡, x2, x3 in JÑ, one has x

i ~x2 = x3

» 3 n G i V . V l í i ¿ 3, «(nat a¡)p~¡ = np¡ = jf¡ o 3 n G A T . V l ^ i ' ^ 3, /!/?,' = n(nat a,)/5j = *,/) *»^l/l0^2/2=^3/3-

Thus *i/i ° * 2 / 2 = (*i* 2 )/ 3 , yielding the required isotopy. D EXERCISES 4A. If (fx,f2,f3): Q -» Q and (g„g 2 , g 3 ): Q -» ß" are quasigroup homotopies, prove that their composite (fxgx, f2g2, f3g3) : Q -* ß" is one too. 4B. Prove that a quasigroup homotopy (fv f2, f3):Q -* P is an isotopy if and only if f¡ bijects for one i in {1,2,3}. 4C. (a) Prove that isotopy is an equivalence relation on any set of quasigroups. (b) Prove that principal isotopy is an equivalence relation on the set of quasigroup structures on a given set. 4D. Prove that isotopic groups are isomorphic. 4E. Prove that a non-empty quasigroup ß is a loop if and only if x/x = y\y for all x, y in Q. 4F. Prove that the multiplication group of the loop (4.4) is a subgroup of the multiplication group of the quasigroup (ß, •, / , \ ). Give an example of a quasigroup (ß, •, / , \ ) for which this subgroup is proper.

LOOPS, NETS, AND ISOTOPY

91

4G. Determine a principal isotope of the quasigroup of (2.4) having 1 as its identity element. Display the multiplication table of the principal isotope. 4H. Determine the right and left divisions in the quasigroup i-JÑ, • ) of (4.6). 4L In the notation of Exercise O 4.4C, consider the 3-net N = (R2, ixy)1 x = yr =» ixy)' = yrxr. Similarly, ixy)r = y'x1. If ß is a loop with identity element 1, then xxl = x1 -x'l = 1 = lx • x' = xxr, so that x1 = xr. Thus in an inverse property loop Q, there is a map J : Q ^ Q; x -* xJ = xr = x1 = x~l with (4.1.1) (*y)'-/*' for *, y in ß. In particular, /?(*)' = Lix1) = L(jc) -1 and Lix)' = Ä(x y ) = -1 Ä(JC) - 1 for x in ß. Moreover, x/y = xy and * \ y = x~ly. An autotopy of a quasigroup (ß, • , / , \ ) is an isotopy ifu / 2 , / 3 ) : (ß, • / . \ ) -* (ß, •. / , \ ) from ß to itself. The set Atp ß of autotopies of ß forms a group under composition (cf. Exercise 4A), containing the automorphism group Aut Q as a subgroup. If ß is an inverse property loop, then there is an action of 5 3 on Atp Q. Proposition 4.1.1. Let Q be an inverse property loop. Then Atp Q is an Syset under the actions (12) : ( / „ f2, f3) -> if{, f{,fj) and ( 2 3 ) : ( / „ / 2 , / 3 ) - * if{, / 3 , f2). Moreover, a bijection appearing in one component of an autotopy appears in the other components in certain other autotopies. Proof. For ifvf2,f3) in Atp ß and x,y in Q, (4.1.1) yields xlf*J -y'W = lf J J J (/'■ -x *y = iy x Y> = ixy)'", so that (//, / / , / / ) lies in Atp Q. Moreover, ifx,f2,hXl2)2 = (fi.fí, fjX.12) - ( / „ / î . / a ) . Now y/» = (x y -xy)^ = J fl Jf J x Hxy) , whence x > ■ yf> = ixy)fi and ( / / , / 3 , / 2 ) G Atp ß . Moreover,

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CHAPTER I GROUPS AND QUASIGROUPS

(/.. / 2 . AX23)2 = (/,', / „ /2X23) = (/„ / 2 , / 3 ). Finally, (/ x , / 2 , /3X(12X23))3 = (/2./3./i/X(12X23))2 = (/3;,/1)/2yX(12X23)) = (/ 1 > / 2 ,/ 3 ). Thus an S3action is obtained (cf. Exercise 3.21). Moreover, if the bijection / , appears in any one component of a certain autotopy a, then it appears in each of the other two components of other autotopies, namely the images of a under various elements of S3. D In view of Proposition 4.1.1, one may define a topomorphism of an inverse property loop Q to be a bijection of Q appearing as a component of an autotopy of Q. The set Tpm ß of topomorphisms of ß forms a subgroup of ß!. An element of ß is a Moufang element if and only if it lies in the orbit of the identity element of ß under the action of Tpm Q. Moufang elements are characterized by the following: Proposition 4.1.2. Let Q be an inverse property loop. The set M = Mfg Q of Moufang elements of Q forms a subloop of Q. For a Moufang element m, L(m)R(m) = R(m)L(m). Moreover, there is a map (4.1.2)

Mfgß -» Atpö ; m ~*

(L(m),R(m),L(m)R(m)),

so that Mit M is a subgroup of Tpm Q. Proof. Let m be a Moufang element of Q, say m = \fx with e At P Q- N o w f o r x i n Q> o n e n a s xfzLim) = lflxfi =xf3, so

(/I.A./S)

that Urn) =ft%

Thus WiMm),Um)-1)

=

(fí\fí\f{'Xfi,hJ2)

= (/i, fi, / 3 ) (/i, / 2 . / 3 ) <23) e Atp Q, whence (Lim)" 1 , L(m)y, / f 1 / / ) = (/f 1 //, L{m\ Um)-1*"* e Atp ß and also (L(/7i), i?(/n), (/f »y/,) = (L(m)-\ L i m V . / f 1 / / ) - 1 e Atpß. Then for x in ß, one has xR(m)L(m) = m xm = (1 • JcX/f 1 ) 7 /! = * ( / f ' V / i = (x • l X / r 1 ) ' / ! = m* • m = xL(m).R(m).Finally, if m and n are Moufang elements, then so are mn = lL(n)L(m) and m" 1 = lL(m) - 1 , so that Af = Mfg ß is a subloop of ß _1

D

An inverse property loop is said to be a Moufang loop if every element is a Moufang element. By Proposition 4.1.2, Moufang loops satisfy the flexible identity (4.1.3)

xy ■ x = x • yx

and the Third Moufang Identity (4.1.4)

zx-yz =

(z-xy)z.

Any group is a Moufang loop. Here are two examples of Moufang loops that are not associative: The first is commutative, so that (4.1.4) may be rewritten in the more readily verifiable form xz • yz = (xy ■ z)z.

LOOPS, NETS, AND ISOTOPY

93

Example 4.13 (Zassenhaus's Commutative Moufang Loop). Let Q be the set Z3. Define a new multiplication ° on Z*, for x = (xu x2, x3, xA) and y = (vi. y2, y3, yA\ by xoy=x+y

(4.1.5)

+ (0,0,0,(JC3 ~

y^)^^-x2y,)).

Then (ß, ° ) is a commutative Moufang loop that is not associative. D Example 4.1.4 (Zorn's Vector-Matrix Algebra). Let M be the set of matrices a

a

b ß

with real scalars a, ß and real 3-vectors a, b. Define the product of

two elements of M by (4.1.6)

a b

a ß

ay + a • d yb + ßd + a X c

y c d S

ac + Sa - b X d b • c + ßS

using the scalar product x • y and cross product x x y of 3-vectors. Define the "determinant" of a matrix by (4.1.7)

a b

a ß

= aß - a -b.

Let Q be the subset of M consisting of matrices whose determinant is 1. Set /= 1 0 0 1 Then (Q, •, I) is a Moufang loop that is neither commutative nor associative. D

There are other ways to characterize Moufang loops. Define the First or Left Moufang Identity (4.1.8)

(zy-z)x

=

z(yzx)

and the Second or Right Moufang Identity (4.1.9)

x(z-yz) = (xz-y)z.

Note that each implies the flexible identity on setting x = 1. The first may be written in the form L(z)L(y)L(z) = L(zy ■ z), involving left multiplications, while the second may be written as R(z)R(y)R(z) = R(z -yz), involving right multiplications. Proposition 4.1.5. In a loop (Q, ■, 1), the three Moufang identities are equivalent. Each implies the inverse property, and thus that Q is a Moufang bop.

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CHAPTER I GROUPS AND QUASIGROUPS

Proof. To begin, suppose that the First Moufang Identity holds. Setting y = z \ 1 yields zx = z((z \ lXzx)), whence x = (z \ 1) • zx. Thus the left inverse property holds, with z' = z \ 1. Using the First Moufang Identity again, (uz)zl = (Z(Z'M) • z)z' = z(z'u • zzl) = z(z'u) = w, so that the right inverse property also holds. By (4.1.1), the map / = / : Q -* Q gives x(z-yz) = ((zJyJ • zJ)xJY = (z'(yJ ■ zJxJ)Y = (JCZ • v)z, so that the Second Moufang Identity holds. Moreover, (zy ■ zXz]x) = z( v • z(z'x)) = z(yx\ so that each L(z) is a topomorphism and each element z = lL(z) is a Moufang element. In particular, the Third Moufang Identity holds. A similar argument shows that the Second Moufang Identity implies the inverse property and the other Moufang identities. Conversely, suppose that the Third Moufang Identity (4.1.4) holds. Setting

z-l/x

yields y(\/x) = (l/*)x-y(\/x) = (il/x)-xyXi/x),

whence y =

(1/x) -xy. Thus the left inverse property holds, with x' — 1/x. Setting v = 1 in (4.1.4) yields the flexible identity (4.1.3). Then zx = zx.z'z = (z.xz')z = z(xz'.z), whence x = xz'.z and the right inverse property holds. Thus Q becomes a Moufang loop: (4.1.4) shows that each L(z) is a topomorphism and each element z = lL(z) is a Moufang element. Finally, since (L(z)J, Uz)R(z), R (z)) = (Uz), R(z), L(z)Ä(z))(23) is an autotopy by Proposition 4.1.1, one has x(z • yz) = (xz • z'Xzy • z) = (z • (jtz)y)y(zy -z) = (xz- v)z, so the Second Moufang Identity also holds. D

EXERCISES 4.1A. Give an example of an inverse property quasigroup with r * I. 4.1B. Give an example of a quasigroup having the right inverse property, but not the left. 4.1C. For a quasigroup Q, prove Aut Q < Atp Q. 4.1D. If Q is an inverse property loop, under what conditions is Aut Q an S3-subset of Atp ß? 4.1E. Let (/j, f2, f}) be an autotopy of an inverse property loop Q. List all 6 elements of the orbit (/j,/^,/^) S3 in the S3-set Atp Q. 4.1F. Show that a Moufang loop satisfies the left and right alternative laws xx y = x ■ xy and y ■ xx = yx ■ x. 4.1G. Verify the claims of Example 4.1.3. 4.1H. Verify the claims of Example 4.1.4. Hints: Show that the "determinant" of a product of vector-matrices is the product of their "determinants". Also show that

a b

a][ ß ß\[-b

-a a

a b

a ß

LOOPS, NETS, AND ISOTOPY

4.11.

4.U.

95

Show that the following conditions on a loop (Q, •, / , \ , 1) are equivalent: (a) Q has the right inverse property; (b) Q satisfies >ct\y = zx\z; (c) Q satisfies yx \ y = x \ 1. Consider a vector-matrix algebra as in Example 4.1.4, but with complex scalars and vectors. Set u, = (1,0,0), u 2 = (0,1,0) and u 3 = (0,0,1). Define 1 en = 0

0 1

i e. = 0

0

e

i

=

0

-u;

for 1 £ 3,

-/

and e

4+j

~

0

IU,

for 1 <>j <. 3.

(U,

(a) Show that Q = {±e ; |0 ^ j' £ 7} forms a 16-element Moufang loop under the vector-matrix product. (b) For 1 £j, k <. 7, show that e¡ek = ±e¡.k, where j • k is calculated in the quasigroup (2.4). 4.1K. (a) (Minkowski spacetime in the vector-matrix algebra.) Show that a 4-vector (f, r) in Minkowski spacetime may be identified with the element y ~' I of the real vector-matrix algebra, so that the "determinant" becomes the Lorentz metric. (Hint: Pick units so that the speed of light c = 1.) (b) (Maxwell's equations in the vector-matrix algebra.) Pick units so that the dielectric constant and the permeability take the value 1. Defining the differential operator matrix d -— D=

dt

V

V d dt

the field matrix F =

0 E+B

-E + B 0

96

CHAPTER I GROUPS AND QUASIGROUPS

and the 4-current J

-P\

show that Maxwell's equations may be written as the single equation DF = / in the vector-matrix algebra. 4.1L. Let ( ß , •, 1) be a group in which each non-identity element has order 3. Define a new multiplication on Q by x ° y = y2xy2. Show that (Q, °, 1) is a commutative Moufang loop. 4.1M. In the multiplication group of a Moufang loop, set P(x) = L(x)~1R(x)~1. Derive the triality relations P(xyx) = P(x)P(y)P(x),

R(xyx) =

R(x)R(y)R(x),

L(xyx)=L(x)L(y)L(x) and P(x)Riy)

1 = P(xy)P(y- ),

P(x)Uy)^P(yx)P(y-1),

R{x)Uy)

= R(xy)R(y-1),

R(x)Piy)

=

R{yx)R(y-i),

L(x)ny)

= L(xy)L(y-1),

L(x)Riy)

=

L(yx)L(y->).

4.2. Loop Isotopes and Bol Loops For a loop (ß, •, 1), the right inverse property may be displayed graphically within the 3-net ß 2 = Net(ß, •, 1) determined by Q. Consider Figure 1.3, in which part of the net is drawn as a sort of optical bench. The vertical line forming the column labeled 1 represents a lens refracting horizontals to diagonals and vice versa. The vertical line forming the column labeled x represents an object mounted on the bench. The point (y, x) in the object has the point (yx, yx \y) in the vertical labeled yx \y as its image. Similarly, the point (z, x) in the object has (zc, zx\z) as its image. In general, the image points of different points of the vertical object line x lie in different vertical lines: The image is unfocused [Fig. 1.3(a)]. However, the right inverse property in the loop ß means precisely (Exercise 4.11) that yx\y = zx\z = x \ 1 = xr. Graphically, this means that the image points of the vertical object line x all lie on a vertical image line x'\ The image is focused (Fig. 1.3(b)). This interpretation of the right inverse property in a loop ß as a property of the corresponding net enables one to specify the stronger property that the loop ß has to possess in order for every loop isotopic to Q also to have the right inverse property. Each vertical z in the net may end up being labeled

LOOPS, NETS, AND ISOTOPY

yx\y

<0

y

Œ>

yx

<£>

CD

<¥

(à)

CE> yx

<£> <£>

zx\z


<3>

zx

97


<J*> zx (*)


Figure U . (a) "Unfocused": yx\y * zx\z. (b) "Focused": right inverse property.

by the identity element of some loop isotope Q' of Q, and the right inverse property in Q' then means that the given vertical z, behaving as a lens, has to focus other object verticals sharply onto an image vertical. As shown in Figure 1.4, this condition reduces to the so-called Second or Right Bol Identity (4.2.1)

x(zy-z) = (xz-y)z.

A loop Q satisfying (4.2.1) is called a (right) Bol loop. Note the subtle distinction from the Second or Right Moufang Identity (4.1.9), namely in the association within the parentheses on the left hand side of the identity. Setting x = 1 in (4.1.9) yields the flexible law, while setting x = 1 in (4.2.1) yields nothing. Thus the Right Moufang Identity implies the Right Bol Identity. On the other hand, the Right Bol Identity is strictly weaker than the Right Moufang Identity:

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CHAPTER I GROUPS AND QUASIGROUPS

yr

z

zyz

Figure 1.4. The Right Bol Identity in a right inverse property loop.

Example 4.2.3. Let ß be the set T\. Define a new multiplication ° on Z32, for x = (*!, x2, x3) and y = (y„ y2, y3), by (4.2.2)

x°y = x + y + (0,0,x 2 y 1 y 2 ).

Then (ß, °) is a right Bol loop that is not Moufang.

D

In the positive direction, the Right Bol Identity implies the right alternative law (cf. Exercise 4.IF) on setting y = 1. Moreover, one has the following characterization. Theorem 4.2.4. A loop is arightBol loop if and only if each of its loop isotopes has therightinverse property. Proof. If each loop isotope of a loop ß has the right inverse property, then Figure 1.4 shows that ß is a right Bol loop. Conversely, let (ß, •, 1) be a right Bol loop, so that (4.2.1) holds. Then xz = x(z(z \ 1) • z) = CczXz \ 1) • z, whence x = (xzXz \ 1). Thus (ß, •, 1) is a right inverse property loop, with z r = z \ 1, and each vertical in Net(ß, •, 1) has the focusing property of Figure 1.4. Let (P, », e) be a loop isotopic to (ß, •, 1). By Theorem 4.5(a), Net (P, o, e ) is isomorphic to Net(ß, •, 1). Thus Net(P, °, e) also has the focussing property, showing that (P, °,e) has the right inverse property. D Corollary 4.2.5. Each loop isotope of arightBol loop is itself arightBol loop. D

Interchanging right and left, i.e. interchanging loops with their opposites, one obtains the First or Left Bol Identity (4.2.3)

(z-yz)x = z(yzx).

LOOPS, NETS, AND ISOTOPY

99

A loop satisfying (4.2.3) is a (left) Bol loop. Thus each loop isotope of a left Bol loop has the left inverse property. Again, the Left Bol Identity is weaker than the Left Moufang Identity. However: Proposition 4.2.6. Let (Q, ■, 1) be a left Bol loop. Then the following conditions are equivalent: (a) (b) (c) (d)

Q is a Moufang loop; Q has the right inverse property; Q satisfies the right alternative law; Q is a right Bol loop.

Proof. If Q is a Moufang loop, then it satisfies the flexible law, and thus is a right Bol loop. In particular, it then has the right inverse property and satisfies the right alternative law. Conversely, it will be shown that each of (b) and (c) implies (a). Since (d) suffices for (b) and (c), the proof will then be complete. (b) =» (a): As a left Bol loop, Q already has the left inverse property. Thus (b) means that Q is an inverse property loop. Setting x = wL(z)~l in (4.2.3) yields that (R(z)L(z\ L(z)~\ L(z)) is an autotopy, so that L(z) is a topomorphism. Thus each element of Q is a Moufang element. (c) => (a): By the right alternative law, (4.2.3) yields (z -yz)z = z(y -zz) = z(yz-z). Substituting yz = x, one obtains the flexible law zx-z — z-xz that suffices to transform the left Bol identity into the left Moufang identity. D

Corollary 4.2.7. Each loop isotope of a Moufang loop is itself a Moufang loop. Proof. A loop is Moufang if and only if it is both a left and a right Bol loop. By Corollary 4.2.5, loop isotopes of left and right Bol loops are themselves left and right Bol loops. G The right Bol identity is strong enough to guarantee the good behavior of exponentiation in a loop. Let (Q, •, 1) be a right Bol loop, so that in particular Q has the right inverse property xyyr=x. Define the integral powers v" of an element y of Q by (4.2.4)

y" =

lR(y)".

Theorem 4.2.8. In a right Bol loop (Q, ■, 1), (4.2.5)

R(y»)=R(y)n

holds. Proof. The equation (4.2.5) will first be proved by induction for non-negative n. It is true for n = 0, n = 1. Then for n > 1, one has

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CHAPTER I GROUPS AND QUASIGROUPS

xR(y") = xiyRiy)"-2 y) = xiyRiy"'2) -y) = x(yy"-2 -y) = (xy -y"-2)y = (xy)R(lR(y)"~2)R(y) = xR(y)n. Thus (4.2.5) holds for all non-negative n. For negative n, one then has R(lR(y)n) = Ä(li?(y r ) |n| ) = /K/) 1 " 1 = R(y)", so that (4.2.5) holds for all integers n. D EXERCISES 4.2A. Verify the assertions of Example 4.2.3. 4.2B. Prove Corollary 4.2.5 equationally, without recourse to nets. 4.2C. Draw and label the analogue of Figure 1.4 corresponding to the left Bol identity in a left inverse property loop. 4.2D. Show that a right Bol loop satisfying (xyY = y V (for xr = l/x) is a Moufang loop. 4.2E. Show that a commutative Bol loop is Moufang. 4.2F. Give an example of an element y of a loop Q such that R(yy) # R(y)2. 4.2G. Let (ß, •, 1) be a group in which each non-identity element has order 5. Define a new multiplication on Q by x ° y = y*xy3. Show that (ß, °, 1) is a right Bol loop. (Cf. Exercise 4.1L.) 4.2H. (a) (The Laplacian in the vector-matrix algebra.) For D as in Exercise 4.1K (b) and for d

_v

— at

show that DD = A •/ with the Laplacian A = d2/dt2 - V2. (b) (Electromagnetic waves in the vector-matrix algebra.) Defining the electromagnetic potential matrix

P=i
A

"A

—
with F = DP, obtain the wave equation A P = /. (Hint: Consider the left inverse property, or the left-handed version of Theorem 4.2.8, in the Moufang loop of invertible elements of the vector-matrix algebra.) 4.3. Right Loops and Loop Transversals In Section 2.2, a quasigroup ß was specified as a set-with-structure (Ö. •. / . \ ) satisfying (2.2.3). This specification breaks up naturally into leftand right-handed parts that share the multiplication. Taking the right-hand

LOOPS, NETS, AND ISOTOPY

101

part alone, a right quasigroup (ß, •, / ) is a set equipped with a binary multiplication • and right division / such that (2.2.3)IR and (2.2.3)SR are satisfied. Recall that (2.2.3)IR yields the injectivity of each right multiplication, while (2.2.3)SR yields its surjectivity. The right division x/y is then expressed in the form xR(y)~l of (2.2.1). Example 43.1. Let ß be a set. Define x.y = x = x/y for x, y in Q. Then (ß> •, / ) is a right quasigroup, with R(y) = 1 for all y in Q. D The extreme Example 4.3.1 shows that the right multiplication map R : Q -» ß! of a right quasigroup ß need not inject. Nevertheless, it is still convenient to study the {combinatorial)rightmultiplication group G, or RMIt (ß, •, / ) , of a right quasigroup (ß, •, / ) , namely the subgroup of ß ! generated by the image R{Q) of the right multiplication map. The right quasigroups of Example 4.3.1 have trivial right multiplication groups. Restriction from right quasigroups to right loops eliminates trivialities such as those inherent in Example 4.3.1. A right loop (ß, •, / , 1) is a right quasigroup (ß, •, / ) with an identity element 1 satisfying (4.3). For a right loop Q, the right multiplication map R : Q -* Q\ injects, since R{x) = R{y) implies x = \R{x) = \R{y) — y. A right quasigroup homomorphism is a set map between right quasigroups that preserves the multiplication and right division. A right loop homomorphism is a right quasigroup homomorphism between right loops mapping the identity of the domain to the identity of the codomain. A right subquasigroup of a right quasigroup (ß, •, / ) is a subset S of ß that is closed under multiplication and right division; a right subloop of a right loop is a right subquasigroup that contains the identity element. A congruence {relation) on a right quasigroup or right loop (ß, ■, / ) is an equivalence relation a on Q that is a right subquasigroup of the direct square right quasigroup (ß 2 , •, / ) with componentwise operations. For a right loop (ß, •, / , 1), define (4.3.1)

{x,y,z)P={x/y)z.

Then {y, y, z)P = {y/y)z = 1 • z = z and (z, y, y)P = {z/y) -y = z for all y, z in ß . In other words, the operation P of (4.3.1) in a right loop ß has the properties of the operation P of (2.2.5) in a quasigroup Q. Consequently, one obtains the following result. Proposition 4.3.2. Let (ß, •, / , 1) be arightloop. (a) The congruence relations on Q are permutable. (b) A right subloop a of Q2 is a congruence if and only if it contains the diagonalrightsubloop Q. Proof, (a) As for Proposition 2.2.3. (b) As for Proposition 2.2.4. D

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CHAPTER I GROUPS AND QUASIGROUPS

Of course, one may also study oppositely-handed versions of the above, namely left quasigroups, left loops, etc. The primary sources of right loops are right transversals to subgroups of groups. Let H be a subgroup of a group (G, •, / , \ , 1), and let T be a right transversal to H in G such that 1 represents H. (Transversals having the identity as the representative for the subgroup are often described as normalized.) Thus G - U /e7 -íft. Define a map e : G -* T; g -* g" by (4.3.2)

jeífe',

so that ge oí ge is the unique representative in T for the right coset of H that contains g. It is also convenient to define a map S : G -» H; g -» gs by (4.3.3)

g-g'g:

Note that I s = V = 1. Moreover hs = h and he = 1 for h in H, while ts = 1 and t' = t for t in T. Now define a binary multiplication * and a binary right division II on T by (4.3.4)

t*u = (tu)e,t\\u =

(t/u)e

for r, w in T, i.e. by íu G H(t * u) and t/u = tu'1 e tf(f||u). Since H(t\\u)u a (i/u) • M = t e. Ht and H(t * M)/M 9 (/n)/u = t e 7/f, one has (r||u)* u = r and (r * w)||u = t for ail f, u in T. Moreover 1 * t = (lf)e = te = t = te = (tï)e = t*l. Summarizing: Proposition 4.33. Let T be a normalized right transversalfroma group G to a subgroup H. Then (T, *, ||, 1) is a right loop. O To within right loop isomorphism, every right loop may be obtained by the construction of Proposition 4.3.3 (cf. Exercise 4.31). Note also that the set bijection T -* H\G;t -» Ht may be used to transfer the right loop structure from the normalized right transversal T to the homogeneous space H\G. In certain circumstances, the right loop of Proposition 4.3.3 will become a loop. If this happens, the normalized right transversal T is called a loop transversal. Thus T is a loop transversal if and only if, for each ordered pair (/, u) of elements of T, the equation (4.3.5)

t*x = u

has a unique solution. The solution x is the result of t dividing u from the left in the loop. Example 4.3.4. Let N be a normal subgroup of the group G. Then a normalized right transversal T from G to TV is a loop transversal. Indeed, the set bijection T -» N\G; t -» M becomes a right loop isomorphism

LOOPS, NETS, AND ISOTOPY

103

iT,*,}\,l)^iN\G,-,/,N) = iG/N,-,/,N), since Nit*u) = Ntu = NNtu = M • r lNtu = Nt-Nu by Corollary 2.4.7. D Example 43.5. Let Q be a quasigroup with element e and multiplication group G. Then the normalized right transversal T = {p(e, v)|v e ¡3} from G to the stabilizer Ge [cf. (2.2.4)] is a loop transversal. Indeed, one may identify T with Q via the mutually inverse maps T -* Q; t -* et and Q -» T; y -» p(e, v). Given s:G -* T; g <-* pie, eg), this yields the right loop iQ, *, e) with x * y = e( p(e, *)* p(e, >0) = ep(e, xp(e, y)) = xpie, y) = xRie \ e)~lRie \ v). This is just the loop iQ, *, e) of (4.4). D Proposition 4.3.6. Let T be a normalized right transversal from a group G to a subgroup H. Then T is a loop transversal if and only if it is a right transversal to each conjugate Hg of H in G. Proof. Suppose first that T is a loop transversal. Note that Hg = g~lHg = (* V ) _ 1 J f c V = H*'- Then for * i n T a n d a in G, a <= Hgx ~ a <E Hgex «» g£ a &Hgs -x «* ige a)s- ige x)e ** ge * x = ig* a)e. Since iT, *) is a loop, there is a unique solution x to the latter equation. Thus T is a right transversal to //* in G. Conversely, suppose that T is a right transversal to each conjugate of H in G. It must be shown that (4.3.5) has a unique solution x. But t * x = u «=» (Or)*? = u <=> Jtfu = ¿/(£c)fi = i/tc <* u e //tt » r" ! u e if'jr. Since J is a right transversal to H', there is a unique x in T for which r - 1 u e H'x, and thus for which t * x = u. D EXERCISES 4.3A. Consider the set iP, • ) of real polynomials with multiplication pix) • qix) = pix) + q'ix) as in Example 2.2.1. (a) Define a right division / on P so that iP, •, /) becomes a right quasigroup. (b) Show that there is no element e of P for which iP, ■, /, e) would be a right loop. 4.3B. (a) Let iQ, •, / ) be a right quasigroup. Show that iQ, / ■ ) is a right quasigroup, with RMlt(ß, / , • ) = RMlt(ß, •, / ) . (b) Give an example of a right quasigroup iQ, ■, /) for which iQ, ■, /) is different from iQ, /, • ). 4.3C. Let 7 = {(1), (123), (132)} and H = {(1), (23)} in the group S3. Determine the right loop iT, *, ||, 1) given by Proposition 4.3.3 in this case. Is it a loop? 4.3D. Let iQ, •, /, \ ) be a quasigroup. Let / / denote the opposite of right division, so that x//y = y/x for x, y in Q. Prove that iQ, \ , //) is a right quasigroup.

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CHAPTER I

GROUPS AND QUASIGROUPS

4.3E. Formulate and prove First Isomorphism Theorems for right quasigroups and right loops. 4.3F. Show that congruence relations on a right quasigroup need not be permutable. 4.3G. Give an example of a right subquasigroup a of the square ß2A of a right quasigroup (Q, •, / ) such that a contains the diagonal Q, but nevertheless fails to be a congruence relation on Q. 4.3H. Suppose that a binary multiplication - is defined on a set Q. Show that there is a right division / yielding a right quasigroup (Q, •, / ) if and only if, for all a, b in Q, there is a unique solution x to the equation xa = b. For finite Q, show that this happens if and only if the multiplication is left cancellative: c-a = da=*c = d for all a,c,d in Q. 4.31. Let (Q, •, / , 1) be a right loop, with right multiplication group G. Let H be the stabilizer of 1 in G. (a) Show that R(Q) is a normalized right transversal to H in G. (b) Let (R(Q), *, II, 1) be the right loop on R(Q) given by Proposition 4.3.3. Show that R:(Q, ■ , / , l ) -► (R(Q), *,||,1) is a right-loop isomorphism. 4.3J. Let T be a normalized right transversal to a subgroup H of a. group G. If T is an Inn G-subset of (G, Inn G), show that (T, *, 1) is a loop. 4.4. Loop Transversal Codes The concept of a loop transversal offers a quick and elementary introduction to the subject known as algebraic coding theory. Algebraic coding theory addresses certain aspects of the problem of transmitting information through channels that are subject to interference. The effect of the interference is to corrupt the signals being transmitted. Nevertheless, algebraic coding theory offers methods of encoding the original information into a signal for transmission, in such a way that the original information may be recovered from a corrupt received signal, or at least so that a signal may be recognized as being corrupt. The information transmission may be taking place through space, sending a message from one physical location to another. On the other hand, it may also be taking place through time, recording a message in a memory and then reading it back later. The usual scheme of algebraic coding theory may be summarized as follows. A finite alphabet A is given, and the set of words to be encoded is the uniform code Ak for some k (cf. Exercise O 4. IB). The information channel carries words from the uniform code A" for some n ~¿.k. The integer n is known as the length of the channel. A subset C of A" is chosen. This subset C is known as the code (or a block code, to avoid confusion with the concept of Definition O 4.1.1, which in turn may be described as a source

LOOPS, NETS, AND ISOTOPV

105

code). The encoding is an embedding Ak -* A" with image C, restricting to a bijection rj:Ak->C. Thus \C\ = \A\k. The integer k is known as the dimension of the code. If a word c from the code C is transmitted through the channel without corruption, then it is received as the same word c. The original encoded word from Ak may then be recovered as cr¡~1. However, the emitted codeword c may have been subject to interference in the channel, being received as a corrupted word x in A". A decoding map (4.4.1)

8 : An -* C

assigns a codeword xs to the received word x. Provided that the received word x was not corrupted excessively from the emitted codeword c, one should expect that xs = c. In particular, one should have cs = c for c in C. Example 4.4.1 (Repetition Codes). Let A = {0,1} and k = 1. Consider a channel length of 3. Define Or) = 000 and ITJ = 111. Thus C = {000, 111). Define the decoding (4.4.1) by 5_1{000} = {000,001,010,100} and S_1{111) = {111, 110,101,011) ("majority vote"). Provided that at most one letter of the emitted codeword gets corrupted in the channel, the decoder is able to recover the codeword. One may extend this scheme to channels of greater odd length. D For further analysis, it is convenient to put an abelian group structure (A, +,0) on the alphabet A. Usually, for \A\ = I, one takes A to be the cyclic group C,. The channel A" is the n-th direct power of A, the i-th projection A" -» A simply selecting the i-th letter of a word (cf. Exercise O 4.1J). Thus the channel A" becomes the abelian group (An, +,0), or more pedantically (A", +.00...0). This abelian group structure may be used to describe the interference taking place in the channel. If an emitted codeword c is received as the corrupted word x, one says that the error x - c was added to c during passage through the channel. The decoder 5 : x -» c is then said to correct the error x — c. To measure the seriousness of the error, one may define the Hamming weight \x\ of a channel word x in A" to be the number of non-zero letters in x. The Hamming distance between two words x, y is then \x — y |. Note that the triangle inequality (4.4.2)

| x + y | < U l + \y\

is satisfied. Indeed, \x + y\ > \x\ + \y\ is impossible, since x + y can only have a non-zero letter in a certain slot if at least one of x and y has a non-zero letter in that slot. Moreover, |JC| = 0 *» x = 0. The decoding may be analyzed using the abelian group structure. An error map (4.4.3)

e : A" -* A"

106

CHAPTER I

GROUPS AND QUASIGROUPS

determines that a received word x was the result of an error xe. Thus x=xs+x£

(4.4.4)

for each x in A". The key idea behind loop transversal codes is the observation that (4.4.4) may just be an instance of (4.3.3). Thus the code C is defined to be linear if it is a subgroup of the channel A". Since A" is abelian, such a subgroup C is normal. As in Example 4.3.4, any normalized right transversal T to C in A" is then a loop transversal. Taking the error map e as in (4.3.2), one obtains the loop transversal T as the set of errors corrected by the code. Note that the loop (T, *,0) defined by (4.3.4) is an abelian group, since the map T -» A"/C; t -» C + t of Example 4.3.4 is a right loop isomorphism of T with the abelian group A"/C. Nevertheless, it is often convenient to continue to refer to the operation * as a loop multiplication, in order to distinguish it from the abelian group operation + on A". Example 4.4.2. Consider the length 3 binary repetition code C of Example 4.4.1. Interpret A as Z2. Then C becomes linear, and the normalized right transversal T = {000,001,010,100} is the set of errors corrected by C. The abelian group multiplication * on T given by (4.3.4) has the table *

000 001 010 100

000 000 001 010 100

010 010 100 000 001

001 001 000 100 010

100 100 010 001 000

Note that the table may be summarized by the specification that the map s:(T, *) -» (A2, +);001 ~ 01,010-» 10,100-» 11 is an abelian group homomorphism. D If one knows a linear code C in a channel A", one may determine a loop transversal T to C by selecting representatives of the various cosets of C. Typically, one picks coset /eaders-representatives having minimal Hamming weight within their cosets. On the other hand, one of the major problems of algebraic coding theory is to determine a suitable code C to begin with, for a given channel A". If the loop (T, *,0) is known, then the code C may be obtained from T by the so-called Principle of Local Duality. To formulate this principle, it is convenient to establish some notation. For elements f1( f2,... of T, define EJlj/, inductively by 0

m

m —1

E í , - 0 and E ' , = 'm+ E ';•

/-i

¿-i

i-i

LOOPS, NETS, AND ISOTOPY

107

Define TllL^t, inductively by

rií, = o and ri',=' m * ru0

m

m-ï

1-1

1-1

1-1

In compound expressions involving loop operations *, II and abelian group operations +, - , the loop operations will bind more strongly than the group operations. For example, t + u — t*u = t + u — (t*u). Proposition 4.43 (Principle of Local Duality). Let T be a loop transversal to a linear code C in a channel A" over a finite abelian group alphabet A. Suppose that T is a set of generators for A". Then

c=

Ef,-ri', <'i

ii-i

ü ^ r

¿-i

Proof. Recall that ts = t for t in T. Induction on m using (4.3.4) then shows that (E?Li'/)
=

£ ', II'. ,-1 ¿-1

('i

'„>Gr

The full force of the Principle of Local Duality comes into play when it is not even known in advance that there is some code C to which a loop (T, *, 0) in A" is transversal. For simplicity, the case A = Z2 = {0,1} will be discussed here. (For a more general version, see Exercise II 2.4T.) Given a channel A", one normally has a list of the errors one would like to correct (e.g. the commonest errors), and this list usually includes the n-element set B of errors of Hamming weight 1. Let T be a 2n_*-element set of errors to be corrected, with T 2 {0} U B. Suppose that T carries a loop structure (T, *, 0) given by an isomorphism (4.4.5)

s:(T,*,0)

-*(A"-k,+,0)

(e.g. as in Example 4.4.2). Let tv...,tm be elements of T. By the closure of ( r , *), the loop product UT-ih always lies in T. On the other hand, the sum EfLji, may only lie in T for certain choices of tv...,tm. The isomorphism (4.4.5) is said to be a partial homomorphism s :(T, +) -* (An~k, +) if (LT-10S = Er-i'/ whenever E¡V, e T. Of course, this means that EJl,*, = n ^ i ' , in such cases, since the two sides of the equation have the same image under the isomorphism (4.4.5).

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CHAPTER I GROUPS AND QUASIGROUPS

Theorem 4.4.4. Let T be a 2"~k-element subset of the length n binary channel An, such that T contains 0 and the n-element set B of errors of Hamming weight 1. Suppose that T carries a loop structure (T, *, 0) given by an isomorphism (4.4.5) such that s :(T, +) -» (A"~k, +) is a partial homomorphism. Then there is a linear code C of dimension k in A" to which (T, *, 0) is a loop transversal. Moreover, T is precisely the set of errors corrected by C. Proof. Note that each element x of A" has a unique expression x = E{6,|i e X] for a subset X of B. Define the syndrome s : A" -* An~k ; £ 6, -» £ bf.

(4.4.6)

ieX

ieX

Since (4.4.5) is a partial homomorphism, it is the restriction of the syndrome to T. Now for x, y in A", with x = Y.ieXb¡ and y = E l 6 y¿„ one has x*+y'=

£*>/ + £&, ? = L{M\i * (X U Y) - (X nY))

ieX

ieY

= (x + y)s.

Thus the syndrome is an abelian group homomorphism. Let C = Ker s be its group kernel s-1{0}. Note that \C\ = \A\k. For x = L¡eXb¡ in A, define xs = I>isXbi - YlieXb¡ and x* = UieXb¡. Then xs e C and x" 6 T, with x=xs+x°. Thus >r = C + 7\ But \A"\ = \C\ ■ \T\, so T is a loop transversal to C in A". Moreover, S : A" -» C; x-> xs and e : A" -* T; x >-* xs surject, indeed e\T = 1 T , so T is precisely the set of errors corrected by C. D Example 4.4.5. For the alphabet Z 2 = {0,1}, consider the set Z\ = {0 = 0000,1 = 0001,2 = 0010 9 = 1001, A = 1010, B = 1011 F = 1111} of hexadecimal digits. This set is to be encoded for transmission through a binary channel in such a way that errors of single Hamming weight may be corrected. Let b¡, for 1 ^ i £ 7, denote the binary word of length 7 and Hamming weight 1 with its unique non-zero letter in the i-th slot. Thus ¿>i = 1000000,..., ft3 = 0010000, etc. Set B = {6,|1 <. i <. 7} and T = {0000000} U B. Define s : T -» 1\ = Z ^ 4 as a partial homomorphism by sending b¡ to the binary representation of i, e.g. b\ = 011. This sets up an isomorphism (4.4.5), e.g. bx * b3 = (b{ + ftps"1 = (001 + 011)s_1 = 010s -1 = b2. By Theorem 4.4.4, the loop transversal (T, *,1) then determines a code C of dimension 4. The 2* hexadecimal digits may be encoded by bijection with C. The elements of C may be determined by the Principle of Local Duality, e.g. bx + fc3 - bx *fe3= 1000000 + 0010000 - 0100000 = 1110000 e C. D

LOOPS, NETS, AND ISOTOPY

109

EXERCISES 4.4A. Describe the repetition code C of length 5 for the alphabet A = {0,1}, by analogy with Example 4.4.1. Decode the received words 10110 and 01010. 4.4B. Taking A as Z2 in Exercise 4.4A, determine a loop transversal T from A5 to C comprising the errors corrected by C. Exhibit an abelian group homomorphism s :(T, *) -» (/I4, +). 4.4C. Use Exercise 4.3J (instead of Example 4.3.4) to show that a normalized right transversal to a linear code is a loop transversal. 4.4D. Write out the cosets of the linear code C of Example 4.4.2, and determine a coset leader for each coset. Are these leaders unique? 4.4E. Write out the cosets of the linear code C of Exercise 4.4B, and determine a coset leader for each coset. Are these leaders unique? 4.4F. For A = Z 3 = {0,1,2}, consider the linear repetition code C = {000,111,222} of length 3. Write out the cosets of the linear code C, and determine a coset leader for each coset. Hence determine a loop transversal T to C. Describe the abelian group (T, * ), either by a table or by an isomorphism with A2. Are the coset leaders unique? If not, repeat the exercise with a different loop transversal U. 4.4G. Verify e\T = lT in the proof of Theorem 4.4.4. 4.4H. (a) Determine all 16 elements of C in Example 4.4.5. (b) Set up an encoding bijection TJ : Z2 -* C. (c) If the hexadecimal digit D is encoded, and subjected to the error ft4 during passage through the channel Z\, show that it may be recovered. (d) If the hexadecimal digit D is encoded, and subjected to the error b2 + b4, to which hexadecimal digit is the received word decoded? 4.41. Consider the binary channel Z25 of length 15. Let B be the set of channel words of Hamming weight 1, and let T be the subset {0} U B of Z2S. By analogy with Example 4.4.5, determine a map s:T -» Z\ = Zl2~n yielding a code C in Z\5 of dimension 11. 4.4J. Consider the binary channel / ¡ o f length 7, as in Example 4.4.5. One wishes to correct errors of Hamming weight 1, and also the so-called burst errors of Hamming weight 2, namely those channel words of weight 2 in which the two non-zero entries are adjacent, e.g. 0011000. Starting with the assignments ¿>j >-» 0001, b2 -» 0010, ¿ 3 -» 0100, b4 -> 1000, b5 -» 1111, b6 i-» 0101 and, ¿>7 -+ 1011, construct a map s : T -> Z\ with domain containing the burst and single-weight errors, by means of which Theorem 4.4.4 builds a code C of dimension 3 correcting the full set T of errors.

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II LINEAR ALGEBRA

1. GENERAL ALGEBRA AND LINEAR ALGEBRA General algebra is the study of sets with structure. Thus its basic objects are sets, and the maps between them are just set functions. Against this background, linear algebra (in its widest sense) may be described as the study of abelian groups with structure. Thus its basic objects are abelian groups, and the maps between them are group homomorphisms. This analogy between general algebra and linear algebra is only one aspect of a very complex relationship. A second aspect is the obvious fact that linear algebra is just a specialized part of general algebra. For instance, linear algebra provides a copious collection of examples of many general algebraic phenomena. A third aspect of the relationship is that linear algebra represents a kind of ideal mathematical world, where calculations are easy and everything is well-behaved. From this point of view, general algebra is often concerned with measuring how far non-linear systems have diverged from the ideal. A major goal of general algebra is then to develop concepts that help to bring non-linear systems back closer to the ideal. In linear algebra, the underlying abelian groups are usually "written additively". In the language of right loops (Section I 4.3), this means that the "multiplication" is addition ( + ), the "right division" is subtraction ( - ) , and the "identity element" is the zero element (0). Ironically, the non-associative operation of subtraction is the most fundamental. Indeed, in an abelian group A, the zero element is a - a for any a in A, while the sum a + ft of two elements a,b is a - ((a — a) - b). Thus a non-empty subset S of an abelian group A is a subgroup iff it is closed under subtraction (cf. Exercise I 2F). The major difference between a set function / : X -* Y and an abelian group homomorphism h : A -» B is that a set function may map any element of X to any element of Y, while an abelian group homomorphism is constrained to map the zero element of A to the zero element of B. The special role of zero elements accounts for much of the distinction between general algebra and linear algebra. For example, recall that the kernel ker/ ill

112

CHAPTER II

LINEAR ALEGEBRA

of the set function / : X -» V is the equivalence relation {(*„ x1)\xlf = x2f) on X. The First Isomorphism Theorem O 3.3.1 factorizes / : X -* Y as the composite of the natural projection nat ker/: X-^X*™1; x>~*xk"f, the isomorphism Xkaf -* Xf; xy"f -»xf, and the injection Xf<-*Y. Forgetting the abelian group structure, this First Isomorphism Theorem for sets applies to the homomorphism h: A -» B. The kernel ker h of h is a binary relation on A. However, it may be described entirely by the group kernel Ker h = {a e A\ah = 0}, a normal subgroup of A. The congruence class * k "'' of an element x of A is then just the coset x + Ker h = {x + a\ah = 0}. Thus Ak"h = A/Keih. The construction of the binary relation ker/ of an arbitrary set function / is a typical general algebraic contrivance that attempts to approximate a simpler construct of linear algebra, the unary group kernel relation Ker h. For abelian groups, the First Isomorphism Theorem may readily be extended. It is customary to write Im h for the image Ah of an abelian group homomorphism h: A -» B. The cokernel Coker A of A is then defined to be the quotient group fl/Im h, usually taken with the natural projection B -* B/lm h; b •-» b + Im h. A sequence ( ~\ 1 \

V1-1/

A

A

§

* ""

Jn-2

.

* An-\

//i-I

A

A

n

"

A

* An + \

Jn + l

* "'

*

A

j

N

of abelian groups and homomorphisms is said to be exact atAn if Im /„_, = Ker/„. It is said to be exact if it is exact at each intermediate group Ax, A2,..., AN_X. The singleton abelian group {0} is often denoted just as 0. The extended First Isomorphism Theorem for Abelian Groups may then be formulated as follows. Proposition 1.1. Let h: A -» B be a homomorphism of abelian groups. Then there is an exact sequence (1.2)

0 — Ker/i—> A-^

B —»Coker A—» 0.

D

Note that the homomorphism 0 -» Ker h is not labeled, since the only possibility sends the unique element 0 of the singleton {0} to the zero element 0 of Ker h. The exactness at Ker h says that the group kernel of Ker h -* A is {0}, or in other words that Ker/i embeds into A. At the other end, the homomorphism Coker h -» 0 is not labeled, since the only possibility sends each element of Coker A to 0 in {0}. The exactness at Coker h says that the image of B -* Coker h is the group kernel of Coker h -» 0, namely all of Coker h. In other words, B -» Coker h surjects. The exactness at A corresponds to the definition of Ker h, while the exactness at B corresponds to the definition of Coker h. If X and Y are sets, then the collection of all set functions f:X-*Y forms a new set Yx. There is a parallel construction in linear algebra. Given two abelian groups A and B, let Hom(/í, B) denote the set of all abelian

GENERAL ALGEBRA AND LINEAR ALGEBRA

113

group homomorphisms h: A -* B. This set is nonempty, since it contains the zero homomorphism 0 : A -* B; a -* 0, i.e. the composite A -» 0 -» B of the unique homomorphisms A -» 0 and 0 -> B. The abelian group structure on B then defines an abelian group structure on Hom(^4, B) via the subtraction (1.3)

- : H o m ( / l > 5 ) 2 — > Uom( A, B);U>S)

>->(f-g:a~af-ag).

For a single abelian group A, the abelian group Hom(A, A) is often denoted as End(y4,+) or just End A. Its elements are called endomorphisms of A. Thus End A is the linear algebra counterpart of the set Xx of maps from a set X into itself. Since composites of homomorphisms are homomorphisms, End A forms a submonoid of the monoid (AA, • ,1). Moreover, there is an abelian group homomorphism j : Z -> End ^4; 1 -> 1. The image of an integer /i under this homomorphism is also denoted n, thus (1.4)

n: A-> A ; a <-> an.

Often one writes «a instead of an, and - a instead of ( - l ) o . EXERCISES 1A. Describe the inversion map (I 1.3.2) of an abelian group A purely in terms of the binary subtraction operation. IB. (a) Prove that an abelian group homomorphism h : A -* B is injective iff Ker/z = 0. (b) Prove that an abelian group homomorphism h : A -* B is surjective iff Coker h = 0. IC. Prove that an abelian group homomorphism h: A -* B is an isomorphism iff the sequence 0-*A—*B-*0is exact. ID. Given an abelian group homomorphism h: A -* B, establish the exact sequence 0 -* Ker h -» A -» A/Kev h -» 0. IE. Let 0 - > / l - > 5 - » C - > 0 b e a n exact sequence of finite abelian groups. Prove that log \A\ - log |B| + log|C| = 0. IF. Let O - ^ o ^ / l j -» ■•• -* An -» 0 be an exact sequence of finite abelian groups. Prove that E?_ 0 (-D'log \A¡\ — 0. IG. For / , g in Hom(yl, B), verify that / - g as defined by (1.3) really is a homomorphism. 1H. Give an example of two group homomorphisms f:G-*H and g:G->H -1 such that f/g : G -» H; x -» (^f) • Ug) is not a group homomorphism. II. Let P and g be quasigroups, with Q entropie. Given quasigroup homomorphisms f:P-*Q and g:P-*Q, define f-g:P-*Q; p -* (pP ■ (pg) and f/g:P^>Q;p" (pf)/(pgl Prove that fg and / / g

114

CHAPTER II LINEAR ALEGEBRA

are quasigroup homomorphisms. Show that the set of all quasigroup homomorphisms / : P -* ß is a quasigroup with • as multiplication. LJ. For an abelian group A, does Aut A form a subgroup of End AI IK. Determine the abelian groups End(Z„, + ) for each positive integer n. IL. Let T be a loop transversal to a linear code C in an abelian group channel A" (cf. Section I 4.4). Exhibit an exact sequence 0 -* C -+ A" -f» T -» 0. IM. Let X,Y,Z be subgroups of an abelian group A or (A, +). Write X - Y = {x - y\x e X, y e Y). Verify the commutative law X - Y = y - X, the associative law

(X-Y)

-Z=X-

(Y-Z),

and the idempotent law

X-X

=X

for subtraction. Conclude that, under subtraction, the set Sb(^4, +) of subgroups of the abelian group (A, +) forms a semilattice. What order relation <, on the set Sb(^4, +) is determined via O 4.3.3 by the semilattice (Sb(y4,+ ), - ) ? 1.1. Products and Coproducts of Abelian Groups Given two abelian groups A, B, their (direct) product is the Cartesian product (O 3.2.1), equipped with the componentwise structure (a,, bx) (a2, b2) = («i - a2, bx — b2\ This structure may also be defined by the middle arrow in the diagram A «-

A XA

A xB

«-

W

AXA

AXBXAXB

•*B

T

■* B

-*

BXB

BXB

an instance of (O 3.2.4). Here the bottom arrows are given by irAxA :(a1,b1,a2,b2) ^* (alta2), and similarly for TTBXB. The relation (O 3.2.4) for products of sets then specializes to an isomorphism (1.1.1)

Horn (C, A x B) s Horn (C, A) x Horn (C, B)

of abelian groups. More generally, given a whole multiset (A¡\i e /> of abelian groups, the product n / e / / l , is the set product with subtraction (and

GENERAL ALGEBRA AND LINEAR ALGEBRA

115

hence abelian group structure) given by the left-hand side of -* A,

UAt TlAiXUAi

♦ K . «■<)

A¡XA¡

in which the bottom arrow projects from the first product to the first factor, and from the second product to the second factor. One then obtains an abelian group isomorphism (1.1.2)

Homftf, n ^ , ) = n H o m ( 5 , A¡). \

iel

I

i<=I

For each i in /, the projection Hom(5, nie¡A¡) -* Hom(B, A¡); f-*fir¡ may be used to verify that Hom(5, n , e / A¡) satisfies the requirements for U¡e,Hom(B,A¡l The group kernel of the projection rrB : A X B -» B; (a, b) -» b is the subgroup A X 0 = {(a, 0) e A X B\a e A). The map LA : A ^ A X B; a -» (a, 0) then yields an exact sequence 0 -» A -2* A x B —^* B -* 0. Similarly, there is an exact sequence 0-*B-^*AxB-^*A-*0. Consider a third abelian group C, the common codomain of group homomorphisms / : A -» C and g : B -* C. Define / ® g : A X B -» C; (a, b) -» af + bg. Then / © g is again a group homomorphism, uniquely specified by the equations tA(f © g) = / and i B (/ © g) = g. Considering the diagram 1A

A

1B

► A x B *

B

f*8 it becomes apparent that the abelian group Ax B with the insertions iA and iB plays a similar role to the disjoint union of sets, or of Af-sets for a monoid M. Taking account of this special role, the product A X B of abelian groups is often written as A © B, and called the (external) direct sum or coproduct of the abelian groups A and B. Since it serves simultaneously as both a product and a coproduct, it may also be described as the biproduct of A and B. (The biproduct is a by-product of the product.) Finally, note that the whole group A © B is generated by the union AiAU BLB of the images of A and B under the insertions iA and iB. The external direct sum A © B of A and 5 is often described as the internal direct sum of the images ALA and 2?ifl. However, it is also customary to identify the direct summands A and B with their images in A © B, thus suppressing the distinction between internal and external direct sums.

116

CHAPTER II

LINEAR ALEGEBRA

The biproduct construction extends to an arbitrary finite number of summands. Let Ax Anbe abelian groups. For 1 <, j] <, n, define n

17: T\A,—* 1-1

J-l

n

TlA¡x

^(ai,...,aj_1,aj+l,...,an)

I I ^(fli.-.-.a,,)

1-1

i-y+i

and n

:A

v i —* TlAi

a

j ~ (o, • • •.o,a y ,o,....0),

1-1

yielding an exact sequence 0 -^Aj

(1.1.3)

X Y\A, - ^ Y\Ai - 0. i+j

(-1

Now consider an abelian group C, the common codomain of homomorphisms fi :A¡-+C for each i. Define fr ® ■■■ © /„ : II".,/I, -» C; ( a „ . . . , a„) -» ¿"_ 1 a,/ i . Then /, ® ••• ®/„ is again a group homomorphism, uniquely specified by the equations t-(/i © ••• ©/„) =// for j = 1,...,/i. Thus the product njLj/l,, equipped with the insertions i¡: A¡ -* n"mlA¡ for y = 1,..., n, becomes the coproduct LI "_ j A¡ of / í , , . . . , An, the analogue of the disjoint union of sets or M-sets. The analogy is expressed by the fact that there is an isomorphism Homl {lA„c\

(1.1.4)

= flHom(^,,C)

of abelian groups. Since the same construction works for both product and coproduct, one often describes the group n"^iA¡ = LI "_ 1^4, as the biproduct ®"_XA¡ of the direct summands Alt..., An. The biproduct is also described as the external direct sum of the A-, or the internal direct sum of the Ajij. It is generated by UMyiyll £j ^ «}, i.e. by E ^ ^ - i y . If a given multiset {A¡\i e. I) of abelian groups is infinite, then the biproduct construction no longer works: the product and coproduct separate. For j e /, define

"7 : FU, -

Il A by

j+iel

16/

ir/iir,. : TI A, ^ A,) = (ir¡ : F U , -» A ) for j ^ i e /. \

;>/6/

/

\

¡e/

/

Define 1- : ¿4y -» n , e / ^ 4 , to yield an exact sequence

(1.1.5)

o ^ ^ n ^ , - n^-o, is/

;>!'€/

GENERAL ALGEBRA AND UNGAR ALGEBRA

117

the analogue of (1.1.3). Define the coproduct LI if=iAito be the subgroup of n¡eIA¡ generated by {A¡i¡\j e /}. If there is an infinite number of non-zero summands A¡, then the coproduct is a proper subgroup of the product Y\is¡A¡. The coproduct lIieiAit equipped with the insertions t¡:A¡-* LI ieIA¡ f° r e a c n j m A possesses the analogue of the defining property for a disjoint union of sets or A/-sets, namely the existence of an isomorphism (1.1.6)

Homf L J ^ . - . c ) s \

í € /

/

is/

YlHom(A¡,C).

Given homomorphisms f¡: A¡ -» C for each i in /, a homomorphism E i e / / i : UiK,A¡ -» C; ati. + - +0,/,,, -» ahfh + ■■■ +a,J,m is specified uniquely by the equations iyŒ,-6//j) = f¡ for y in /. The coproduct llie,Aj is also known as the (external) direct sum of the A¡, and as the internal direct sum

of the AJLJ.

EXERCISES LIA. Verify that (1.1.1) and (1.1.2) really do give isomorphisms of abelian groups. LIB. Let A be an abelian group. (a) Verify that subtraction gives a homomorphism A XA —*A. (b) With the diagonal A : A -> A X A; a -» (a, a), show that there is an exact sequence 0 -» A — * A x A —*A-*0. 1.1C. Prove that an abelian group C is the internal direct sum of subgroups A and B if and only if C = A + B and A n B = {0}. j p LID. Let 0 -* A —* E —*g-»0bean exact sequence of abelian groups. (Such an exact sequence, with three inside groups sandwiched between zero groups, is called a short exact sequence. Then E is called an extension of A by Q.) Prove that the following three conditions are equivalent : (a) The surjection p: E -* Q has a section s : Q -» E that is a homomorphism; (b) The injection j : A -* E has a retraction r : £ -» A that is a homomorphism; (c) E is the internal direct sum of Im j and Cokery. (If these conditions obtain, then the extension and the exact sequence are described as split.) LIE. Prove Hom(A © B, C) s Hom(yl, C) X Hom(5, C). 1.1F. (a) Let Av..., An be finite sets. Prove that \\ú'U\A¡\= ï-'l-MX (b) Let Au . . . , An be finite abelian groups. Prove that iog|e;. 1 >i,.|=E;'. 1 iogi^.|.

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1.1G. Verify that (1.1.4) and (1.1.6) really do give isomorphisms of abelian groups. 1.1H. For abelian groups A and B, can you exhibit the disjoint union A U B as a subset of the underlying set of A ©5? 1.11. Prove that an abelian group A is the internal direct sum of a finite set {Al An) of subgroups iff the following two conditions are satisfied: (a) A = L^A,; Q>) VI *j*n, A, n La,A¡-{0). 1.1J. Let A be a finite abelian group. Prove that A is the internal direct sum of its Sylov subgroups. 1.1K. Let A, Av..., Anbe abelian groups. Prove that A is the biproduct of Au...,A„ if and only if there are homomorphisms p¡: A -* A¡ and jj : Aj^> A for each i such that: (a) VI ^h,i<,n,jhpi= 8hi\A¡; (b) 1A = Z'LiPüi in HomU, A). 1.1L. Let jf be a set, and let A be an abelian group. Define an abelian group structure on Ax by x(f - g) = xf - xg. For x in X, define the evaluation map irx : Ax -> A; f -> xf. Prove that Ax, together with the evaluation maps, becomes the direct product of |.Y| copies of the group A. 1.1M. An element / of Ax (as in Exercise 1.1L) is almost zero (or "has finite support") if {x e X\xf # 0} is finite. For an element x of X, define the insertion ix : A -* Ax; a >-* (y -> if y = x then a else 0). Prove that the set Ax of almost zero functions, together with the insertions, becomes the coproduct of lA'l copies of the group A. 1.1N. Continuing the notation of Exercises 1.1L, and 1.1M, let XK be the submonoid of (Ix, +,0) generated by {8x\x e f l , where 8X is the delta function li,. Prove that XK, together with the injection i : X -> XK; x -» 8X [cf. (I 1.5.1)], becomes the free commutative monoid on X. For m e XK and x e X, prove that mirx is the multiplicity of occurrence of x in the multiset m. 1.2. Matrices A key feature of linear algebra is the use of matrices to describe homomorphisms between biproducts of abelian groups. Given abelian groups Ax,..., A,,Bu..., Bm, the isomorphisms (1.1.4) and (1.1.2) yield a compos-

GENERAL ALGEBRA AND LINEAR ALGEBRA

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ite isomorphism

(

m

/

\

l

© A„ © B,; s

i-i

;-i

7

I

m

© Horn A¡, © 5.

/-i l

= ©

i-l

m

\

y-i

Im

©Hom(/l(.,5y);/~ E

;-l

E/y

í-l y-i

with (1.2.1)

/y-i^eHom^,^.).

It is convenient to display the various / i ; of (1.2.1) in an array: /n (1.2.2)

'''

f\j

'''

fin

/n

fij

Jim

fn

fu

f,im

[JiJllxm

[J>j\•

This array is the matrix of /. The map f¡j is the (i, j)-component or (/, j)-entry of the matrix. The element [£i .../•, . . . / i m ] or ( / i , . . . , / 7 , . . . , / i m ) of rTyliHomM,-, 5 ; ) is the i-th row of the matrix. The element

/« ///

or (fljt..., f¡¡ f,¡) of n'_iHom(y4,, B¡) is the j-th column of the matrix. A matrix such as (1.2.2), with / rows and m columns, is described as an / X m-matrix. If / = m and A¡ = B¡ for 1 ^ i ¿ /, the matrix is square. Example 1.2.1. The IXm matrix of ir.-iy e Hom(©i_jy4, ®"_x A) has the identity map 1 e Hom(^4, ^4) as its (i,))-entry, and the zero map 0 e Hom(^4, A) as its (/', /)-entry for ï # i or / =£ ;'. This matrix is known as the (i, j)-th elementary matrix E'lm or E''. D Example 1.2.2. The matrix of the identity map 1 e Homi©),^,, ©j_,/l7) has the identity map 1 e Hom(y4(, A¡) as its (/, i)-entry for 1 <. i ^ /, and the

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zero map 0 e Hom(^4„ A¡) as its (i, ;)-entry for i ¥= j . This square matrix is known as the / X / identity matrix I¡ or /. D Example 1.2.3. The matrix of the zero map 0 e Hom(©|_j/l(, ©", l B¡) has the zero map 0 e Hom(A¡, B¡) as its (i, ;')-entry for 1 <. i ^ / and 1 <,'} <, m. This matrix is known as the / X m zero matrix 0/Xm or 0. D Example 1.2.4. The group (Z, 2 , +) is the internal direct sum Z 4 © Z 3 of its non-trivial Sylov subgroups (cf. Exercise 1.1 J), via insertions t, : Z 4 -♦ Z 12 ; x<*> „ 9^; t. -3 "^ 2 12 ; v<3> -» 4y<12> and projections vi :Z 12 -» Z 4 ; <12> 4 z<">~z< \ir2 -» z<3>. For integers r and n with n > 1, 12 -» Z 3 ; z consider the abelian group endomorphism r:(Z„, +) -» (Z n , +); x<'1> -* (a) [cf. (1.4)]. What is the matrix of 5 : Z12 -» Z,2? For JC<4> in Z 4 , equation (1.2.1) yields x<4>5n = x<4>t,5ir1 = (45x)<4> = *<4> and x<4>512 = x<4>ij57r2 = (45*)<3> = 0. Similarly, for y<3> in Z 3 , one obtains y<3>52, = y<3>i25ir1 = (20y)<4> = 0 and >><3> 522 =y<3>i25ir2 = (20y)<3> = 2y<3>. Thus the matrix of 5 is f¿ °l. D Example 1.2.5. For A = (Z 2 , +), let C be a linear code of dimension k in the binary channel A" = ©"_,/! of length n, given by the construction of Theorem I 4.4.4. As noted in the proof of Theorem I 4.4.4, the syndrome s: ©"_, A -* ©"_~j A of (4.4.6) is an abelian group homomorphism. Its n X (n - A:)-matrix is known as the parity check matrix of the code C. For 1 <. i <. n, the insertion i¡:A-* © " . ^ maps 1 to the binary word b¡ of length n and Hamming weight 1 with its unique non-zero letter in the i-th slot. Thus the i-th row of the parity check matrix is the binary word bf of length n — k. For example, the parity check matrix of the code of Example 4.4.5 is 0 0 0 1 1 1 1

0 1 1 0 0 1 1

1 0 1 0 1 0 1

Now consider further abelian groups Cl m

n

Horn ©2?,, 0 C J =

©

C„ and the isomorphism

© H o m ( 5 , , Q ) ; g ~ E E 8jk-

For 1 ^ i: ^ / and I ^k <.n, the (i, AO-component of the matrix of

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121

fg G Hom(©¿_j A¡, ©£_, Ck) is given via (1.2.1) as

(

m

\

m

m

E ^j\girk = E lifTTjLjgTTk = £ /yfy [for the third equality, cf. Exercise 1.1 K(b)]. Summarizing, {jg)lk-

(1-2.3)

m

Lfijgjk-

7-1

This is the equation by means of which the / X /n-matrix of / is multiplied by the m X «-matrix of g to yield the / X «-matrix of fg. EXERCISES 1.2A. Determine the (square) matrix of irki¡ G Homí©"..^, ©"_, A). 1.2B. Give an example of a 2 X 2-matrix that is not square. 1.2C. Writing (Z45, +) as the internal direct sum Z9 © Z5 of its non-trivial Sylov subgroups (cf. Exercise 1.1J and Example 1.2.4), determine the 2 x 2 matrix of the homomorphism 13 : (Z45, +) -> (Z45, +); z<45> -> (13z)<45>. 1.2D. Determine the parity check matrix of the code of Exercise I 4.41. 1.2E. Derive a formula for the product E''E'''' of two elementary square / X /-matrices E'', E''ï as in Example 1.2.1. 1.2F. Consider the elements of Horn (Z 2 ©Z2,Z2©Z 2 ) represented by the matrices / 2 ,[° ; ] , [ • ¡ ] , [ ¡ ?],[¿ }],[} }]. Prove that they form a group under multiplication that is isomorphic with 5 3 . (Cf. Section O 2.) 1.2G. A square matrix (1.2.2) is symmetric if f¡j = f¡¡ for all i, j . Is the product of symmetric matrices symmetric? 1.2H. A square matrix (1.2.2) is diagonal if f¡¡ = 0 for i # j . Which endomorphisms of (Z12, +), considered as endomorphisms of Z 4 © Z 3 , have diagonal matrices? 1.21. Let A be a non-trivial abelian group. For T G Sn, define n

n

i-l

¡-I

pT = E h^iT G End © -A. (a) Prove that p:Sn-* End ©"_, A; T •-» pT is a monoid homomorphism. (b) The matrices of the pT are called permutation matrices. Prove that an « X « square matrix is a permutation matrix if and only if it has 1A as the unique non-zero entry in each row and each column.

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1.2J.

Let (Z„, •, / , \ ) be a quasigroup on the set Zn. With notation as in Exercise 1.21, and A = Z„, prove that the matrix of E ^ ' P R O ) IS a Latin square that yields the left division table when bordered by 0 , 1 , . . . , « - 1 in order. 1.2K. For i e R , define

exp0==[ r

cos

f

sin

L -sinO

^ e R i L.

coso]

Show that the equation exp 6 exp

(x-y)z

=

xz-yz,

a consequence of the definition of - and the fact that z is an endomorphism. On the other side, they are connected by the left distributive law (1.3.2)

x(y-z) =xy-xz,

a consequence of the definition of subtraction. In compound expressions such as the right hand sides of (1.3.1) and (1.3.2), the multiplication binds more strongly than addition and subtraction. Abstracting the properties of the set of éndomorphisms of an abelian group, one obtains the following definitions. Definition 13.1. A ring, or unitalring,orring-with-a-one,or Z-algebra, is a set R equipped with an abelian group structure and a monoid structure, connected by the right and left distributive laws. D Definition 13.2. A ring, or non-unital ring, orring-without-one,is a set R equipped with an abelian group structure and a semigroup structure, connected by the right and left distributive laws. D The ambiguity implicit in these definitions—does a ring have a multiplicative identity or not?—is a perennial source of confusion. As a very vague and unreliable rule of thumb, it is more natural to consider rings with one when focussing on individual rings and ways of constructing them from other individual rings, while rings without one tend to arise naturally when studying

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123

whole classes of rings together. In practice, one needs to determine from the context whether the word "ring" is referring to a unital or non-unital ring. Example 1.3.3. Let A be an abelian group. Then (EndA, +,-,1^) is a unital ring. Defining x * y = 0 for x, y in A, one obtains a non-unital ring (A, + , *). Rings of such form (A, + *) for an abelian group (A,+ ) are known as zero rings. D Example 1.3.4. Let B be a set. Define the operation xor of symmetric difference or exclusive or on the power set £P{B) by X xor Y = {X - Y) U (Y - X). Then {3»{B\ xor, n , B) is a unital ring. Let J8«. U(J5) be the set of finite subsets of B. Then(^8< a(B), xor, n ) is a non-unital ring. D

A ring R is commutative if the semigroup i? is commutative. The rings of Example 1.3.4 are commutative. On the other hand, End(Z © Z) is not commutative, since (ir,i2X7r2ii) = u - ,^, while (fl^Xw^ij) = ir2t2. Example 1.3.5. The set Z of integers forms a unital ring under addition and multiplication. The set 2Z of even integers forms a non-unital ring under addition and multiplication. D A set function f:R-> S between unital rings is a {unital) ring homomorphism if it is a homomorphism both of abelian groups and of monoids, for the respective parts of the ring structure of R and 5. Similarly, a set function between non-unital rings is a {non-unital) ring homomorphism if it is both an abelian group homomorphism and a semigroup homomorphism. A ring isomorphism is a bijective ring homomorphism. Rings R and S are isomorphic, written R = S, if there is a ring isomorphism between them. A subset R of a unital ring 5 is a {unital) subring if it is both an abelian subgroup and a submonoid. A subset R of a non-unital ring S is a {non-unital) subring if it is both a subgroup and a subsemigroup. Thus 2Z is a non-unital subring of Z, but not a unital subring. As a trickier example, {&K{0,1}), xor, n , {0,1}) is a unital ring that is a non-unital subring of the unital ring {&>{N), xor, n , H), but not a unital subring. Now consider a ring (5, + • ). For each element s of 5, there are various elements of Ss determined by s:

Í

R+{s) L{s) R{s)

:S

->S;x->x+s; :S-*S;x~*sx; :S -> S;x^xs.

Thus R+{s) is the right (or left) "multiplication" by s in the quasigroup (5, +). In the present context, it is called translation by s. The map L{s) is

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called left multiplication by s. The left distributive law says that it is an endomorphism of (S, + ). Similarly, the right distributive law says that the right multiplication R(s) is an endomorphism of (S, + ). The right multiplications feature in the linear algebraic version of the general algebraic Theorem 0 4.3. Theorem 1.3.6. Let (S, +, •, 1) be a unital ring. Then R:S-* End(S, +); s <-* R(s) gives a ring isomorphism S -* SR of (5, + • ) with a ring ofendomorphisms of the abelian group (S, + ). Proof Theorem O 4.3 shows that R gives a monoid isomorphism. The left distributive law xR(s - t) = x[R(s) - R(t)] shows that R also gives an abelian group isomorphism, and hence a ring isomorphism. D The map R of Theorem 1.3.6 is called the right regular representation of the ring S (cf. Section I 1). Given a family <5,|i e / > of rings, the Cartesian product YlieIS¡, along with the projections TT¡ : n i e / S , -» Sjt yields a product of rings, since it is simultaneously the product of abelian groups and monoids or semigroups. Specifically, the ir; are ring homomorphisms, and given ring homomorphisms f¡:T -* S¡ with common domain for each i, there is a unique ring homomorphism f'-T-* n , e / 5 ( with fvj = f¡ for each j . This works equally well for unital as for non-unital rings and homomorphisms. Theorem 1.3.6 and the product construction for rings suggest consideration of matrices with ring elements as entries. Let (S, +, ■ ) be a (non-unital) ring. For positive integers l,m, let 5™ denote the set of / X m arrays or matrices (1.2.2) with entries from S. For 1 <, i < I and 1 ^j^m, the map TT¡j : S" -» S; f1-* fj sending a matrix to its (i, y)-entry is a projection making S¡" the product of Im copies of the ring S. Thus SJ" has componentwise ring structure. The componentwise product of two matrices is called their Hadamard product. If S is a unital ring, then S¡" becomes a unital ring with the "all-ones" matrix / or /

J,m= E

m

¡-i

LEO,

¡-i

as identity. Now for an additional positive integer n, define a matrix product (1.3.4)

SrxSnm^S?;(f,g)~fg

=

E fijgjk

j-i

as in (1.2.3). In particular, the abelian group S} of / x / square matrices becomes a ring under the matrix product. If S is unital, then S¡ is unital, with /, as identity element. In this case Theorem 1.3.6 extends to show that

( ©5, es l

m

\

•-i

y-i

/

l

m

;[/ i ; ]~ E LHfu) ¿-i y - i

GENERAL ALGEBRA AND LINEAR ALGEBRA

125

is an abelian group embedding, and a unital ring embedding if / = m. For an element / of a non-unital ring S, define the / x I scalar matrix f = [/)7]/x/ by f¡j = if Í = j then / else 0. Then there is a ring embedding (1.3.6)

S-»S/;/~/.

For scalar matrices, the matrix product and Hadamard product coincide. EXERCISES 1.3A. Let (5, +, -, 1) be a unital ring. (a) Under the sum x®y=x+y-l and the product x°y = x + y — xy, verify that (S, © , °, 0) is a unital ring. What is its zero element? (b) Writing ' : S -* S; x -» 1 - x, verify the De Morgan Law (xy)' = x' o y'. (c) Interpret ' in the unital ring of Example 1.3.4. 1.3B. Let (5, +, • ) be a non-unital ring. Define a product on the abelian group Z © S by (m + sXn + t) = mn + sn + mt + st [cf. (1.4)]. Prove that (Z © S, +, •, 1) is a unital ring, and that the abelian group insertion i s : 5 -» Z © 5 is a non-unital ring homomorphism. 13C. Prove that every non-unital ring is isomorphic with a ring of endomorphisms of an abelian group. Contrast with Exercise O 4L 13D. Let <5,-|i e /> be a multiset of non-unital rings. Prove that the group coproduct L I i e / S ( forms a subring of the product ring Ylie¡S¡. Does this result extend to unital rings? 1JE. Let 5 be a commutative ring. Define the trace tr : Sj -» S; [f¡j] -» E'_i/¡¡. Prove that the trace is an abelian group homomorphism, and that tK/g) = trig/"). O F . Let 5 be a unital ring. Show that there is a unique unital ring homomorphism Z -* S. U G . (a) Prove that (1.3.4) defines an action of the monoid S™ on the set 5J". (b) If S is unital, show that the action is faithful. (c) Give an example of a ring S and a positive integer m for which the action is not faithful. (d) Show that the m x m permutation matrices [cf. Exercise 1.21 (b)] form a subgroup 5 m of the monoid 5™, for S unital. (e) Let X = {^i^ll <. j £ m) be the set of elementary 1 X m-matrices. Show that X is an Sm-subset of the 5m-set S¡".

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(f) Prove that the trace of a permutation matrix is the number of fixed points it has in X. O H . Let S be a ring and / a positive integer. An / x /-matrix [/ y ] in S\ is diagonal if ftj = 0 for i ¥= j (cf. Exercise 1.2H). Prove that the set of I X I diagonal matrices with matrix product forms a subring of the ring S¡, under matrix product, and a subring of the ring Sj under Hadamard product. O I . Let 5 be a unital ring. Prove that an / X / diagonal matrix [/ y ] is invertible in the monoid (S1,, •, /,)—cf. Exercise 11.3M—iff each of its diagonal entries f¡¡ (for 1 £ i £ /) is invertible in the monoid (5, •, 1). 1.3J. Let S be a non-trivial unital ring. Set X = {1,2,..., n). For a relation p on X, the incidence matrix Ap of p is the n X n matrix E (j , ;)ep E¿ ; n . Verify the following: (a) AXxX

= /;

(b) Ax = /; (c) EuApAaEJ1 * 0 =» O'.y) e p °
1.4. Ideals, Fields, and Domains A subset K of a semigroup S is a sink if VA: e iC, Vs e S, ks e AT and sk e K. If / : 5 -» T is a non-unital ring homomorphism, then the group kernel Ker/ is a non-unital subring of the ring S and a sink in the semigroup 5. This follows since {0} is a sink in the semigroup T. In general, a non-unital subring Kofa ring S is said to be an ideal of 5, written K < S, if K is a sink in the semigroup 5. This means that the quotient group S/K becomes a ring under the well-defined semigroup product (1.4.1)

(x + K)(y+K)

=xy + K.

If S is unital, then S/K is unital with identity element 1 + K. In any case, K is the kernel of the natural projection R+(K): S -* S/K; * -» x + K, and the exact sequence 0 -» K -> S -* S/Ä" -» 0 of groups becomes an exact sequence of non-unital rings. Each ring S possesses the trivial ideal {0} and the improper ideal S. Note that no proper ideal of a unital ring S can contain the identity element. A ring S is simple if it has no proper, non-trivial ideals. Let Id S denote the set of ideals of a ring 5. The set is partially ordered by containment. If {K¡\i e /} is a set of ideals of 5, then the intersection n , e / £ , is also an ideal. For a subset X of 5, the ideal (X) or (X)s generated by X is C\{K < S\X Q K). For a finite subset X = {xv..., xr), write (X) = <Xj,..., xr). If 5 is commutative and unital, then (x) = xS for x in 5. In any ring S, (x) is called the principal ideal generated by the element x of S.

GENERAL ALGEBRA AND LINEAR ALGEBRA

127

Let 5 be a unital ring. Denote the group of units or invertible elements of the monoid S by S*, i.e. l e i ' « 3x~l e S. xx'1 = 1 = x~lx. If S* = S {0}, then the ring 5 is a skewfield or division ring, and a field if it is commutative. Proposition 1.4.1. Let S be a commutative, unital ring. Then S is a field if and only if it is simple and non-trivial Proof. Let x be a non-zero element of simple, non-trivial S. Then xS, as a non-trivial ideal of S, contains the identity element, i.e. 3x~l e S. xx~l = 1 = x~lx. Conversely, suppose each non-zero element is invertible. Then {0} < K < S, say 0 # x e K, implies 1 = xx~x e XS Ç K, whence K = 5 and S is simple. D Corollary 1.4.2. A non-zero ring homomorphism, whose domain is a field, injects. Proof. For S a field and 0 # / : S -» T a ring homomorphism, the group kernel of / is a proper ideal of S. Thus Ker/ = {0}, i.e. / injects. D Let Prop 5 denote the set of proper ideals of a non-trivial ring S. Thus (Props, Q) is a non-empty poset. Since the union U i e / ^ ( of a chain {Kj\i e /} of ideals is an ideal, the poset is inductive. Then by Zorn's Lemma, the ring S has maximal (proper) ideals. Indeed, a similar argument shows that any given proper ideal K of S is contained in a maximal ideal of S. Write Max S for the set of maximal ideals of S. Proposition 1.4.3. Let S be a non-trivial, commutative unital ring. For K <S, K e Max S «=» S/K is a field. Proof. K is maximal «=» S/K is non-trivial and simple (cf. Exercise 1.4J) «=> S/K is a field. D A commutative, unital ring S is a field iff S — {0} is a subgroup of the monoid S. Analogously, one has the following; Definition 1.4.4. Let 5 be a commutative, unital ring. Then S is a domain or integral domain iff S — {0} is a submonoid of the monoid S. D Note that a field is certainly a domain. On the other hand, the ring Z of integers is an integral domain (whence the name) which is not a field. By Proposition 1.4.3, an ideal M in a commutative, unital ring S is maximal iff S/M is a field. One thus defines an ideal P in a commutative, unital ring S to be prime iff S/P is a domain. The spectrum of S is the set Spec 5 of prime ideals of S. [The rationale for this terminology becomes apparent later

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(Exercise 3.4L).] Since fields are domains, maximal ideals are prime. On the other hand, {0} is a prime ideal of Z which is not maximal (being contained in the ideal 2Z, for example). For a positive integer n, the ideal nZ is prime «=> Z/nZ = Z„ is an integral domain «=> n is prime. In general, prime ideals P in a commutative, unital ring S are characterized by each of the following equivalent conditions:

(1.4.2)

P prime *» S/P is a domain «• (VJC, y e S, x £ P and y «È P => xy «È P) » (VJC, y e S, jqy e P => x G P or j i e P ) .

EXERCISES 1.4A. Show that the product (1.4.1) is well-defined. 1.4B. Verify the claims of the second paragraph. 1.4C. Determine the group of units of the ring Z M . 1.4D. Determine the group of units of the ring (Z 2 )| of 2 X 2 matrices over the ring Z 2 (cf. Exercise 1.2F). 1.4E. Formulate the First Isomorphism Theorem for rings. 1.4F. If S is a commutative, non-unital ring, show that the principal ideal generated by an element x is xTL + xS. 1.4G. If S is a unital, non-commutative ring, show that the principal ideal generated by an element x is SxS •■= {Lr¡^iS¡xt¡\r e N; s¡, t, e S). 1.4H. If 5 is a non-unital, non-commutative ring, show that the principal ideal generated by an element x is xZ + xS + Sx + SxS, the latter summand as in Exercise 1.4G. 1.41. Prove {K,\i G /} ç Id S => E, e / K¡ < S. 1.4J. For K < S, prove that there is a bijection {/ < S\K ç /} -* Id(S/K); J -> J/K. 1.4K. (a) For ideals K, L of a commutative ring S, prove KL ■■= \ E k,lt\r e N; k¡ e K; I, e l\ < S. (b) Prove KLçKnL. (c) For K, Le Spec(Z), prove KL = K n L. (d) Give an example of ideals X, L of a commutative ring 5 with KL c ATnL. 1.4L. For positive integers m,n, prove m divides n iff nZ ç mZ.

VECTOR SPACES AND MODULES

129

1.4M. A non-zero element x of a ring 5 is a zero divisor if there is a non-zero element y of S with yx = 0 or xy = 0. If 5 is unital and commutative, prove that S is an integral domain iff it has no zero divisors. 1.4N. Let F be a field. For n > 1, show that the matrix ring Fnn is a simple ring with zero divisors. (Hint: show that a non-zero ideal contains an elementary matrix. Show that no proper ideal can contain an elementary matrix.) 1.40. For a prime number p, show that the zero ring having Z p as the underlying abelian group is simple. 1.4P. Let S =

a b

-b a,beZ, a

(a) Prove that 5 is a subring of the ring (Z3)l of 2 x 2 matrices over the ring Z 3 . (b) Prove that 5 is a field. 1.4Q. Let / be an interval in U. Let C(7) be the set of continuous functions f: I -* U, considered as a ring under pointwise addition and multiplication. For x in /, show that Mx = {/ e C(I)\xf = 0} is a maximal ideal of C(7). 1.4R. Let (S,|i e /> be a family of non-unital rings. Prove that the group coproduct LI i e / S , is an ideal in the product ring n , e / S , . 1.4S. Let / : / ? - » S be a commutative, unital ring homomorphism. Show that there is a well-defined map / _ 1 : Spec S -* Spec R; P -» f~l(P). 2. VECTOR SPACES AND MODULES The general-algebraic concept of a right 5-set X for a monoid 5 was defined in (I 1.1) via the representation R:S -* Xx, a monoid homomorphism. Given a ring S, the linear-algebraic concept of a right S-module X or Xs or (AT, +,S) is defined analogously: it is an abelian group X, together with a ring homomorphism (representation) (2.1)

R:S ^> End X; s->

(x^xs).

If S and R are non-unital, the module X is non-unital. If S and R are unital, the module X is unital. Note that a unital 5-module X is an 5-set for the monoid S. A right S-module X is faithful if the representation (2.1) injects. For a faithful module, one often identifies elements of the ring S with their images under the representation R. If 5 is a field, Corollary 1.4.2 shows that a non-trivial unital S-module X is faithful. Unital modules over a field S are

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known as vector spaces over the field. The elements of 5 are called scalars in this context. The image s or R(s) : X -* X of a scalar s under the representation is called a homothety or scalar multiplication by the scalar s. This terminology is occasionally extended to modules over more general rings 5. Example 2.1. For a unital ring 5, the right regular representation of Theorem 1.3.6 is a representation in the sense of (2.1), making 5 a unital right 5-module. For reasons that become clear later, this module is described as the free monogenic (right) S-module. D Example 2.2. For a ring S and a positive integer m, the action (1.3.4) of the monoid 5™ on the set 5j" of row matrices [cf. Exercise 1.3G(a)] makes the abelian group S" a right 5™-module, faithful if S is unital [cf. Exercise 1.3G(b)]. D Example 23. Let B be a subset of a set A. Then the semigroup action of the semilattice (&>(B), n ) on the set g"(A) given in Exercise 11A makes the abelian group (&>(A), xor) a right module over the ring (9>(B), xor, n ) of Example 1.3.4. D Most of the general algebraic constructions for monoid actions studied in Section I 1 have their counterparts in linear algebra. For instance, given a ring homomorphism f:S-*T and a right T-module X with representation R:T -» End X, the composite fR:S-* End A" is a representation making X a right 5-module. Example 2.4. For a ring 5 and a positive integer m, the ring homomorphism (1.3.6) mapping 5 to the scalar matrices makes a right 5-module 5f out of the right 5™-module of Example 2.2. The scalar multiplications in the right 5-module 5J" are "entrywise" : (su..., sm)s = (s,s,..., sms). In analogy with the terminology of Example 2.1, the right 5-module 5¡" is described as the free (right) S-module on m generators. D Given two right 5-modules (X, +,S) and (Y, +,S), a right S-module homomorphism f:(X,+,S)-+(Y,+,S) is an abelian group homomorphism / : (X, + ) -> (Y, + ) and an 5-homomorphism / : (X, 5) -» (Y, 5). If the context is clearly linear-algebraic—5 a ring and X,Y modules—an S-module homomorphism may just be described as an 5-homomorphism. Denote the set of 5-homomorphisms from X to Y by S(X,Y). Now S(X,Y) is a subgroup of Horn (X, Y). If 5 is commutative, then S(X, Y) is an 5-subset of the 5-set Yx with s : Yx -» Yx; / -» / s for s e 5. Thus S(X, Y) is itself an 5-module in this case. In an 5-module (X, + ,S), an 5-invariant subgroup K of X is a submodule of (X, +, 5), a relationship denoted by K <. X (and K <X for proper containment). Corestriction of the representation R:S-* End X to a

VECTOR SPACES AND MODULES

131

representation S -» End K makes K an S-module and the embedding K «-» X an S-homomorphism. Given a submodule K of AT, the group quotient X/K is an S-module via (x + K)s = xs + K for x e X, s e S. Then (X/K, S) is known as the quotient module. Given an S-homomorphism f:(X, +,S) -» (y, +,S), the group kernel Ker/ is a submodule of X and the image Xf = Im/ is a submodule of Y. It is customary to carry over the exact sequence notation and terminology from groups to modules. Thus an S-homomorphism / : X -» Y yields an exact sequence (2.2)

0 — » K e r / — * X - ^ Y — ► Coker/—» 0,

the analogue of (1.2). In particular, (2.2) defines the cokernel Coker/ as the quotient Y/ Im /. Given a family (X¡\i e /> of S-modules, the Cartesian product n¡eíX¡ is the product both of the groups X¡ and the S-sets X¡. In other words, given S-homomorphisms f¡:Y -» X¡ with common domain Y, there is a unique S-homomorphism f:Y-* T\ie¡X¡ with /TT, = / for each i. Now the abelian group coproduct \lielX¡ is an S-subset of the S-set Y\ie¡X¡, and thus an S-module. It is the coproduct of the S-modules X¡: Given S-homomorphisms / :X¡-*Y with common codomain Y, there is a unique S-homomorphism f:\limIXi-*Y with iif — fi for each i. If there are only finitely many non-trivial modules X¡, then the product and coproduct coincide to give the biproduct (X, +, S) to X from the free monogenic S-module with lL(x) = *. (b) Show that x -» L(x) and p >-* \p are mutually inverse maps yielding an abelian group isomorphism X s S(S, X). Under what conditions does this become a module isomorphism? 2B. Verify that the action (x + K)s = xs + K (for x e X, s e S) on the quotient A'/X of an S-module A' by a submodule K is well-defined. 2C. Formulate and prove a First Isomorphism Theorem for S-Modules (over a given ring S). 2D. Let (A', +, S) be a unital S-module (over a unital ring S). Let (Y, +, S) be a non-unital S-module, and let f'.{X, +,S) —» (y, +,S) be an Smodule homomorphism. Show that Im / is a unital S-module inside the non-unital S-module Y.

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CHAPTER II LINEAR ALEGEBRA

2E. Let A and B be abelian groups. (a) Show that A is a Z-module under the representation j : Z -* End A of (1.4). (b) Show that an abelian group homomorphism / : A -* B is a Z-homomorphism. 2F. How to make a module unital. Let A be a non-unital module over a non-unital ring 5, with representation R:S-> End X. Let U : Z -» End A" be the unique unital ring homomorphism of Exercise 1.3F. Let U + R : Z © 5 -» End A" be the abelian group coproduct of the maps U and R. Show that, if Z © 5 is the ring of Exercise 1.3B, then U + R is a representation making X a unital module over the unital ring Z © 5. 2G. How to make a module faithful. Let X be an 5-module that is not necessarily faithful, with representation R : S -» End X. Define the annihilator An X or Ans A to be the group kernel Ker R in 5. Factorize R:S -» End A through R : 5 / An A -» End A using the First Isomorphism Theorem for Rings. Show that A is a faithful 5/An A-module. 2H. For a family <X¡\i e / ) of 5-modules, exhibit abelian group isomorphisms

sfy.n^-nsiY,^) v

¿6/

and

'

S(UX»Y)* V

i e /

/

¿e/

Y\s(xitY) is/

for a given 5-module Y. 21. Let {/ be a subring of a ring 5. Show that the concatenation of the embedding U *-» 5 with the right regular representation 5 -* End 5 of 5 makes 5 a right {/-module. 2J. Let A be an 5-module. Let Sb(A; +, 5) or SbAs denote the set of 5-submodules of A. Show that (SbAs, +), with K + L = {k + l\k e K, I € L], is a semilattice, and a monoid with identity element {0}. 2K. (a) Let / : Y -» Z be an 5-module homomorphism. For an 5-module X, show that there is an abelian group homomorphism R(f) = S(X, f) : S(X, Y) - 5( A, Z); 6 -» 6f. (b) Given a second 5-module homomorphism g:Z -* W, show that S(X,f)S(X,g) = S(X,fg). (c) If Y" and Z are unital, show that there is a commutative diagram Y s S(S,Y)

f

S(S,f)

Z s S(S, Z) whose rows are the isomorphisms of Exercise 2A(b).

VECTOR SPACES AND MODULES

133

2.1. Duality and Transposed Matrices Let 5 be a non-commutative ring, with abelian group (S, + ) and semigroup (5, • ) connected by the right and left distributive laws (1.3.1) and (1.3.2). Then the opposite ring Sop is the ring with abelian group (S, 4-) and semigroup (5,°) opposite to (S, • ), as in (I 1.4). The right distributive law for S yields the left distributive law for Sop. A left S-module SX or (X, S, + ) is a right Sop-module Xs<* or (X, +, Sop). The scalar multiplications in a left S-module are usually written on the left of their arguments, following (I 1.5) and (I 1.6). Example 2.1.1. Let S be a ring. Then 5 is a right S-module under the right regular representation R:S-* End(S, + ); s -» (x -* xR(s) = xs) and a left S-module under the left regular representation L : S"9 -» End(S, + ); s -* (x -* xL(s) = sx). O Example 2.1.2. For a ring S and a positive integer m, the action (1.3.4) of the monoid S™ on the set Sxm of column matrices (cf. Example 2.2) makes S¿, a left S™-module. D Given left S-modules (X, S, + ) and (Y, S, + ), a left S-module homomorphism f :(X, S, +) -* (Y, S, + ) is a right Sop-module homomorphism f:(X, +,S o p ) -» (y, +,S o p ). Thus / is an abelian group homomorphism, and s(xf) = (sx)f holds for s in S and x in X. If the ring S is commutative, then there is no distinction between right and left S-modules. In this case the scalar multiplications may be written on either side of their arguments. For an abelian group {X, +) and right S-module (Y, + , S ) the abelian group Homi-X", Y) of abelian group homomorphisms f:X-*Y has the structure of a right S-module, with scalar multiplications defined by (2.1.1)

x(ft) = (xf)t

for x e X, f e Hom(X, Ys)s,t G S, via the right S-module structure of Y. One sometimes refers to the definition (2.1.1) as a mixed associative law: an associative law because of its form, but mixed because its arguments are from the various sets X, Hom(^T,y), and S in order. Now suppose that (X, +, S) is a right S-module, while (Y,+ ) is just an abelian group. Then Hom(.Y, Y) has the structure of a left S-module, with scalar multiplications defined by the mixed associative law (2.1.2)

x(sf) = (xs)f

for x e X, s G S,fesHom(Xs,Y) via the right S-module structure of X. Note the verification that Hom(Z, Y) is a left S-set: x((siS2)f) = U(sj.s 2 ))/ = ((xsj)^)/ = ( X Ï J X Î Z / ) = xdsiisj)) for oc e * , s, e S, / e Hom(AT, Y). Given a right S-module homomorphism f:U-*V,a left S-module homomorphism L(f) = Hom(/, y ) : Hom(F, Y) -► Hom(í/, Y); 6 ~ fO is defined.

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CHAPTER II LINEAR ALEGEBRA

Indeed, u(s(fO)) = (us)fO = duf)s)9 = ( U / X J 0 ) = uifisd)) for u e U, s e 5,0 e Horn (K, Y). The definition may be displayed by the following commutative diagram:

u

(2.1.3)

¡e*

*v

Hom(/,y)

1 e

Given a second right 5-module homomorphism g : V -» W, one obtains (2.1.4)

Hom(g,Y)Hom(/,Y) = Hom(/g,Y).

Indeed, for

be a family of right 5-modules. By (1.1.6), there is an abelian group isomorphism 8 :Hom(LI i e / j4 ( , G) -» n, e / Hom(y4„ G), under which the image (L¡eIf¡)6 of an element Lielf¡ : UieIAt -» G is determined by its projections (Eje/^flw) =./} : -<4; -» G for y G /. Now for Í in 5, one has (sZie¡f¡)0irj = Œ, ei s/i)ôir ; = sf¡, whence (sE l 6 //j)ô = s(CLie¡f¡)d), so that 0 becomes a left 5-module isomorphism. In other words, the dual ( LI, e / A¡)T of the coproduct is (isomorphic to) the product TlieIAj of the duals. D Example 2.1.5. Let / be a finite interval in U, and let C(/) be the set of continuous functions f: I -* R. Then C(I) is an R-module under ( / - gXx) - fix) - g(x) and (rfXx) = rf(x) for x e /, r e R, and f,gs C{I\ (Cf.

VECTOR SPACES AND MODULES

135

Exercise 1.4Q.) Let G be the abelian group R. Then for g e CU), the Riemann integral Jg : CU) -» R; / •-* f,f(x)g(x) dx is an element of the dual C(/) r . Moreover, the map CU) -* C(/) r ; g -* Jg injects (Exercise 2.11). For a point y in /, the Dirac S-function 8y is not a function I -* U, let alone a continuous function, but it determines an element Jy : CU) -» R; / •-» / ( v) of the dual C(/) r . It is thus convenient to define fjf(x)8y(x)dx =/(y), and to think of Sy as a "generalized function" or "distribution" 8y: I -> M.. D For right 5-modules Au...,A,,Bu...,Bm, consider an abelian group homomorphism / e Hom(©'_j ^4,, ©;™, 5¿). Recall that / has a matrix [ij/i)], XB as in (1.2.2). By Example 2.1.4, one obtains (®i_, / l , ) r = H o m ( © ( ' _ , . 4 „ G) = Hom(LIÍ_i<4,. G) s n ! . , H o m ( / J f , G ) = ©'_! HornU,, G) = ®/_j /4[ and ( ©;™j ß ; ) r s €T 1 5/", where the isomorphisms are left 5-module isomorphisms. Thus / , a priori an element of Hom((© ; 1 1 £ ; ) r , i © ' . ! ^ ) 7 " ) , becomes an element of Hom(© ; 1, 2?/, ©¡_1 ^4f ). Two questions then arise: what is the m X I matrix of fT, and what relationship does it bear to the I X m matrix of / ? To answer these questions, consider /3- e Z^7" for 1 <, j <>m in the following commutative diagram: ®Ah

hfyßj

B<

t-i

A-l

f"jßj

ßj-

"jßj

Then under the composite

BJ-U

r ©ß[-U

k-l

h-\

®ATh-Z*Aj,

the element jS, maps to i, /TT, /3y = / i ; /3, = ty/;7/. Thus: Theorem 2.1.6. / / / e H o m J ^ . , ^ , ^ . , 5,) /uw I X m-matrix [/y],XM, tften tfte ¿«a/ homomorphism fT e Horn! ©™ j 2?/", ©'_, y4[) has an mxl ; matrix whose (j, i)-entry is the dual ft of the (i, j)-entry of the matrix off. D The m X I matrix

/Ä

/£

/Í"

/5

f7"

f¡¡

fl

fT

J im

Jim

136

CHAPTER II

LINEAR ALEGEBRA

is called the transpose '/ii

hi

■

flm

fn

fij

■

Jim

fn

fij

■

flm

of the matrix (1.2.2). EXERCISES 2.1A. Let 5 be a unital ring. Show that the left regular representation L-.S0? -> End(S, +) of Example 2.1.1 makes 5 a faithful unital left 5-module, i.e. a faithful unital right 5op-module. /

g

2.1B. Let 0 —► A —► B —► C —► 0 be an exact sequence of right 5-modules over a ring S. (a) Show that, for an abelian group G, the sequence Hom(A,G)


is exact. /

g

(b) Give an example of an exact sequence 0 —► A —» B —* C —» 0 and a ring S for which 0 — Uom(A,G)

^ ^ -

Hom(5,G) ^ ^ -

is not exact. /

Hom(C,G) «— 0

g

(c) Show that, if 0 —» A —> B —» C —* 0 is a split exact sequence, then so is 0 «— H o m ^ . G ) ^ ^ i Hom(B,G) ¿ = ^ £ 1 Hom(C,G) «— 0. 2.1C. Let G be the abelian group / . Let m be an integer. Determine the dual mT of the Z-homomorphism m : Z -» Z; x >-* xm. 2.1D. Let X and Y be (right) modules over a commutative ring 5. Show that S{X, Y) is a submodule of the (right) S-module Hom(.Y, Ys) given by (2.1.1) and of the (left) 5-module Horn (Xs, Y) given by (2.1.2).

VECTOR SPACES AND MODULES

137

2.1E. (a) Let SX be a left 5-module over a ring 5. For an abelian group G, write elements / of HomíA-, G) on the left of their arguments, i.e. / : Ar -» G; x -» /(*). Show that Hom(A, G) is a right 5-module T via the action specified by the mixed associative law (fsXx) = SX f(sx) with / e XT, s e 5, x e X (b) For a right 5-module X, show that evaluation gives a map A" -* A"Tr; x <-* ( / -* JC/"). For what structures on A and XTT is this map a homomorphism? 2.1F. For a ring 5, let G be the abelian group (5, +). For a positive integer m, show that matrix multiplication 5™ X 5¿, -> 5} = 5 as in (1.3.4) yields a left 5™-module homomorphism from the left 5™-module 5¿, of Example 2.1.2 to the dual (S?)T of the right 5£-module S? of Example 2.2. 2.1G. Consider the abelian group Z„ as a Z-module. Let G be the circle group, the group {z e C|zz = 1} of complex numbers of unit modulus, under multiplication. Show that Z„ is isomorphic with its dual lJn. 2.1H. Let y be a point of the interval / of Example 2.1.5. Define the Heaviside function Hy : / -» R by // y (x) = if x < y then 0 else 1. (a) Show that Z/^,, while not itself an element of C(7), determines an element f~ ¡,f{x)Hy(x)dx = f,my^f(x)dx of C(/) r . (b) If formal "integration by parts" is to hold, how should the "derivative" H'y be defined? 2.11. Prove that the function C(7) -» C(/) r ; g -> / g of Example 2.1.5 injects. 2.2. Solving Linear Equations Let X and Y be abelian groups, and let / e HomiX, Y). For a fixed element _y of y, an equation of the form (2.2.1)

xf = y

with x e A ' i s called an c/jî/ie equation or an (inhomogeneous) linear equation. The equation (2.2.2)

xf=0

is then called the corresponding homogeneous equation. In general, an equation of the form (2.2.2) is called a linear equation or a homogeneous linear equation. Of the alternative terminologies, the former is more correct algebraically, while the latter is more widely used. The term y of (2.2.1), especially if it is non-zero, is called the inhomogeneous term of (2.2.1). Classically, given / and y in (2.2.1), one is interested in determining the solution set f~l{y). The endeavor is known as solving (2.2.1). The key issues

138

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are the dual notions of existence and uniqueness of solutions, i.e. the respective truth values [ y e I m / ] and [Ker/={0}]. If solutions exist, i.e. y e Im/, then any element x of f~l{y) is called a particular solution of (2.2.1). The coset x + Ker/ is called the set of general solutions of (2.2.1). Thus a general solution of the inhomogeneous linear equation (2.2.1) is the sum of a particular solution and a solution to the corresponding homogeneous equation (Exercise 2.2B). Note that the homogeneous equation (2.2.2) always has at least one solution, viz. 0. There are three possibilities for (2.2.1): (0) (1) ! (°°)

no solutions — y € Im / ; a unique solution - y G Im / and Ker/ = {0} ; many solutions - y e Im / and Ker / > {0}.

The equation (2.2.1) is described as inconsistent or overdetermined in case (0), well-posed in case (1), and underdetermined in case (<*). Example 2.2.1. Consider the syndrome homomorphism s : (A", + ) -» (A"~k, +) of the proof of Theorem I 4.4.4. Suppose that a word w has been received. The decoding problem may be construed as an example of solving the inhomogeneous equation xs = ws for x. Since (I 4.4.5) surjects, the equation has a particular solution ws e T. The codewords, as elements of Ker s, are solutions to the corresponding homogeneous equation xs = 0. The solution x of the inhomogeneous equation xs = ws chosen by the decoder is then the sum w = we + ws of the particular solution we and the solution ws of the corresponding homogeneous equation. D Example 2.2.2. A vehicle suspension system maintains a variable vertical displacement s between an axle and a mounting point on the body of the vehicle. The suspension system includes two key components: a spring and a damper. The spring exerts a net force ks proportional to the displacement *, while the damper exerts a force Is' proportional to the vertical velocity s' (the time derivative of s) of the body with respect to the axle. The suspension system has to control an inertial force due to vertical movement of the body that is given as ms" by Newton's Second Law, where the constant m is proportional to the mass of the body. If the axle is subjected to a net vertical force of F (the effect of the weight of the vehicle having been subtracted both from this term and from the spring force ks), then the behavior of the suspension system is described by the differential equation (2.2.4)

ks + ls' + ms" = F.

Let X be the abelian group C2(0,1) of real-valued twice-differentiable functions s : (0,1) -* R from the time interval (0,1), with pointwise addition.

VECTOR SPACES AND MODULES

139

Let y be the abelian group R<0, !) of real-valued functions on (0,1), again with pointwise addition. Then the function / : X -* Y; s -» ks + Is' + ms" is an abelian group homomorphism, and the differential equation (2.2.4) becomes an instance of (2.2.1). It is customary to check the performance of a suspension system (e.g. to see whether the damper constant Z is still good) on a stationary vehicle by displacing the body, and then releasing it at time 0. Over the time interval (0,1), one is then observing a solution to the homogeneous equation corresponding to (2.2.4). The validity of this procedure as an indication of the dynamic behavior of the system, described by a solution to the inhomogeneous equation (2.2.4), rests on the fact that this solution is the sum of a particular solution to (2.2.4)—dependent on F, and thus on the irregularities of the surface over which the vehicle is traveling—and of a solution to the homogeneous equation that is likely to be similar to the solution observed during the stationary test. D Now let 5 be a unital ring. By virtue of Theorem 1.3.6, 5 is identified with its image under the right regular representation. Given positive integers / and m, a system of m linear equations in I unknowns is an equation of the form (2.2.5)

xA = b

with x = (xv..., x,) e S[, A = [atj] e S,m, and b = ( 6 „ . . . , bm) e 5J". The components of x are the unknowns, and A is called the coefficient matrix of the system. By (1.3.5), the coefficient matrix A in S" determines an element of Hom(S¡,5im), while b determines an element of Horn (S\,S?). One identifies matrices with their images under (1.3.5). Thus (2.2.5) becomes an instance of (2.2.1). The (/ + 1) X m-matrix of A © b e Hom(5{ © S\, S,m) = Hom(5{+1,5J"), i.e. the matrix A with b added as an extra bottom row, is called the augmented matrix of the system (2.2.5). Note that the augmented matrix describes the system (2.2.5) completely. Two systems xA = b, xA0 = b0 of m linear equations in / unknowns are equivalent if their solution sets agree, i.e. J 4 _ 1 ( ¿ ) = A¡¡l(b0) as subsets of S{. The general technique for solving (2.2.5) involves passage from the original system, through a series of intermediate equivalent systems, to a final equivalent system whose solution set is immediately apparent. Such a final system is described as reduced, while the passage to a reduced system is described as reduction. During the reduction, it suffices to record the augmented matrices of the intermediate systems, since a system xA = b is determined completely by its augmented matrix A © b. Reduction is achieved by post-multiplying (/ + 1) x m augmented matrices by invertible m X m square matrices. Proposition 2.2.3. If there is an invertible matrix E in S™ with {A © b)E = A0 © b0, then xA = b is equivalent to xA0 = b0.

140

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Proof. Firstly, note {A © b)E = A0 © b0 implies (A0 + b0)E~l = A © b, so the relationship between the two systems is symmetrical. Then x e A~l(b) ç S[ =» xA = b => xA0 = xAE = xbE = xb0 => x eA^1^) cS{,i.e. ^ - 1 ( ¿ ) 1 1 l çAo (b0). Similarly, Aô ^) QA~ (b\ whence equality of the solution sets. D To perform a reduction in easy steps, one selects particular kinds of invertible matrices: permutation matrices (cf. Exercise 1.21), diagonal matrices with invertible entries (cf. Exercises 1.2H, 1.3H), and matrices of the form / + sE'> for i # j and s e 5, where E'J is the (i, ;')-th elementary matrix (cf. Example 1.2.1). Consider a system (2.2.5) of 3 equations in 2 unknowns, i.e. =b

l

*1«11 + *2«21

xian +x2a22 = b2

(2.2.6)

x a

+*2 f l 23

\K

=

b

3

Permutation. Post-multiplying the augmented matrix of the system by the permutation matrix p(123) yields *13 *22

*23

0 0 1

1 0" 0 1 == a 23 0 0

"12

*21

an b.

In general, post-multiplication of an augmented matrix by a permutation matrix pT yields the augmented matrix of a new system such that the i-th equation of the old system becomes the iT-th equation of the new system. The i-th column of the old matrix becomes the »T-th column of the new matrix. The matrix is said to have undergone the elementary column operation of permutation. Rescaling. Post-multiplying the augmented matrix of (2.2.6) by the invertible diagonal matrix Cl

0 0

0 c2 0

0 0 c3

yields All «21

¿1

a12 Û22

b2

«13 ' Û23

b

K

'ci

0 0

0 c2 0

0' 0 = c3

anc1 a21c1 Vi

al2c2

a c

fl

«23^3

C

22 2

b2c2

n3

*3C3

In general, post-multiplication of a matrix by a diagonal matrix with unit i-th diagonal entry c, yields a new matrix whose i-th column is the i-th column of

VECTOR SPACES AND MODULES

141

the old matrix postmultiplied by the scalar c¡. The matrix is said to have undergone the elementary column operation ofrescaling. Shear. Post-multiplying the augmented matrix of (2.2.6) by the invertible matrix / + sE12 yields oil «21

¿>

an a22 b2

«13 «23

b,

1 0 0

s 1 0

0 un 0 = «21 1 *1

ans + an

«13

a2\S + a22

«23

¿>!* + b2

¿>3

In general, post-multiplication of an augmented matrix by a matrix / + sE'' with i ¥= j postmultiplies the i-th equation of the system by s and adds the result to the y-th equation. The matrix itself, having had its i-th column post-multiplied by s and added to its y'-th column, is said to have undergone the elementary column operation of shear. The name comes from the geometric action Q •-* /~7 of / + sE21 on the Cartesian plane R2. An elementary column operation is of one of these three kinds: permutation, rescaling, or shear. The submonoid of S% generated by the elementary column operations is called the monoid of column operations. If the unital ring S is a field, then the reduction and ultimate solution of a system (2.2.5) may be achieved by the method known in the West as Gaussian elimination (but already known to the Chinese of the Former Han dynasty two millenia before Gauss' time). Consider a system over the field IR of real numbers with augmented matrix 1 1 1 1 1

2 0 2 0 2

1 1 1 1 1

Initially, the entries of the coefficient matrix are described as unused, and are not underlined. An invertible entry of the coefficient matrix is then selected as a so-called pivot, and underlined: 1 1 1 1 1

2 0 2 0 2

1 1 1 1 1

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By rescaling the column of the pivot, the pivot element becomes 1 : 1 -1 1 -1 1

2 0 2 0 2

1 1 1 1 0 1 .0 1

0 1 2

0

1 -1 1 -1 1

0" 0 = 1

1 0 1 0 1

1 1 1 1 1

Using shears, the other elements of the row of the pivot are cut down to zero, and all elements of the row and column of the pivot element in the coefficient matrix are underlined and considered as used: 1 0 1 0 1

1 1 1

] 1 ] 1 ] 0 1 L 1 _

0 1 0

=

1 -1 1 -1 1

1 0 1 0 1

l" 1 1 1 -1 0 1 1.

=

0 -1 0 -1 0

1 0 1 0 1

0" 1 0 1 0

0" -1 1.

() 0" ' 1 1L 0 0 () 1 0

0 1 0

0" -1 1

At the next stage, an unused invertible entry of the coefficient matrix is chosen as pivot and rescaled to 1. Again, the other elements of the row of this new pivot are cut down to zero with shears (during which process no zero elements of rows of earlier pivots ever become non-zero), and at the end the elements of the row and column of the current pivot in the coefficient matrix are underlined and considered as used: 0 1 0 1 0

1 0 1 0 1

0 1 "l 0 0 0 1 1 1 0 0

0' 0 = 1

0 0 0 0 0

1 0 0 1 1 0 0 1 1 0

VECTOR SPACES AND MODULES

143

This step is repeated until, as now in the example, no unused non-zero elements remain in the coefficient matrix. An / x m-matrix is said to be column-reduced of column rank r if it has the form 0rx(m-r) for some 0 <, r <, mint«, m] and an arbitrary (/ — r) X r-matrix B(t_r)Xr. The final step in Gaussian elimination is a permutation to bring the coefficient matrix into column-reduced form: 0 0 0 0 0

1 0 1 0 1

0] 1 0 1

oj

0 1 0

0 0 1

1" 0 = 0

1 0 0 0 1 0 1 0 0 0 1 0 1 0 0

If the augmented matrix is not column-reduced, then the system (2.2.5) is inconsistent, since one of the (r + l)-st,..., m-th columns of the final augmented matrix Ä © b' corresponds to an equation xfi + ••• +xfi = b'j with non-zero b'j. Otherwise, one may pick xr+ x,...,xm to be arbitrary members of the field S (so the system is under-determined if r < m), and then use the first r equations of the reduced system to obtain x,,...,xr in terms of b\,...,b'r and possibly xr+1,...,xm. In the example, one obtains xx = 1 — x3 and x2= -x4. Thus the solution set is (1,0,0,0) + (-1,0,1,0)R + (0,-1,0,

EXERCISES 2.2A. (a) Show that the solution set of (2.2.1) is always closed under the operation P of (12.2.5). [Hint: For / e Hom(X, Y) and a, b, c e X, one has (a, b, c)Pf = (of, bf, cf)P.] (b) Show that the solution set of (2.2.1) is a subgroup of X iff the equation is homogeneous. 2.2B. If pf = y in (2.2.1), prove p + Ker/ - {* e X\xf = y). 2.2C. Give an example of abelian groups X, Y, an element j e Y , and / e HomíA', Y) such that (2.2.1) has exactly two solutions. 2.2D. Let C2[0,27r] be the ring of twice-differentiable functions a :[0,2TT] -+ U, with pointwise operations. (a) Show that C 2 [0,2ir] -> Ri0-2*!; a -» a" + a is an abelian group homomorphism. (b) Show that a = x3 — 6x is a particular solution of a" + a = x3. (c) Find the general solution of a" + a = x3.

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2.2E. For elements / and y of a field, consider the equation xf = y. Show that the truth values of uniqueness and existence of solutions are given respectively by [Ker / = {0}] = [/=/= 0] and [y e Im / ] = [ / = 0 l y = 0] + [ / * 0 ] . 2.2F. Solve the systems of linear equations over R with the following augmented matrices: 1 2 1 3 5 - 3 7 14 1 3 0 -3 1

3 9 3 15 -1 2 -3 4 -4

2 1 1 » 2. 0 --1 1 --1 3

1 1 1 3

2 1 -2 1

1 2 -3 1

-1 -1 3 1

0" -7 3 . 1 -3

tern of linear equations over Z with augm ente« "5 2 0 2

21 9 5 0

7" 3 2 1

2.2H. A system (2.2.5) of linear equations over a field S has 5 solutions. Show that S is isomorphic to Z s . 23 Bases and Free Modules Let 5 be a unital ring, and let V or Vs be a unital right 5 - module. Consider the left S-module S or SS. As in Exercise 2.1E, the group Hom(S, V) of abelian group homomorphisms 6 : S -» V; r -» 8(r) becomes a right S-module Hom(S, V)s with actions specified by the mixed associative law (0iXr) = 9(sr) for 0 e Hom(S, V) and r,se.S. The group S{S, V) or S(SS, Vs) of right S-module homomorphisms p:S-+ K[cf. Exercise 2A(a)] is a subgroup of Horn (S, K). Moreover, it is right S-submodule S(SS, Vs)s, since for p e S(S, V) and u, s, f e S one has (psX/t) = p(srt) = p(sr)f = (psXr)t. The middle equality holds since p is an S-homomorphism, while the other two are instances of the mixed associative law. As in Exercise 2A(b), there are mutually inverse abelian group homomorphisms L(l) : S(S, V) -* V; p -* p(l) and L : V -> S(S, V); v -» (Uv) : r ~ vr). Now for p e S{S, V) and s e S, one has L(l) : pi >-» ps(l) = p(sl) = p(li) = p(l)s, so that L(l) —and hence also its inverse L—are S-homomorphisms. Thus for any unital right S-module V, the modules V and S(S, V) are isomorphic. It is often convenient to identify them.

VECTOR SPACES AND MODULES

145

For an index set /, an I-linear combination is an element of L I i e / S = S(S, LI ¡ejS). One usually realizes the coproduct L I i e / 5 as the module So of almost-zero functions / -» S (cf. Exercise 1.1 M). In particular, if I—{!,..., m], then L I i e / 5 is the module 5J" of 1 X m matrices, as in Example 2.4. For a right 5-module V, an I-tuple of elements of V is an element of the product (cf. Exercise 1.1L) V = Uie,Vs n , e / 5 ( 5 , F ) = S(UieIS,V), the latter isomorphism as in Exercise 2H. Composition of S-homomorphisms yields a map (2.3.1) sis, Us) X s[US,v) V

16/ '

V

IS/

'

- S(S,V) sV;(í,f)

~ if.

Example 23.1. For a finite index set / = {1,..., m), an /-linear combination is called an m-linear combination. In particular, the element ( 1 , 1 , . . . , 1) e Sm = LI T-\s « usually written as 2 or ££.,. Then for v = (u, vm) e Km = F ; s 5(11 ,"1,5, K), (2.3.1) specializes to (E, «) -» Ew = E^i"/- D In general, given ^ e 5Q and f:I-*V, one has £/ = E /f „ 0 '/* 'Í- The sum is finite since i is almost zero. Note also ( 0 -» 5X0 -♦ F) = 0. Now consider a fixed /-tuple / : / - » K For such / , (2.3.1) yields an 5-homomorphism (2.3.2)

R(f):sls,Us)->V;€~ff. X

iel '

Properties of the /-tuple / : / -* V, or of the multiset (cf. Exercise 2.3A), are defined by properties of the map (2.3.2). Definition 23.2. The /-tuple / : / -» V or multiset {if\i e />: (S) spans (V) if R(f) surjects; (I) is independent if R(f) injects; (B) is a basis if /?(/) bijects. □ (The alliteration is a useful mnemonic.) Note that / : / -» V is independent, i.e. R(f) injects, iff Ker/?(/) = 0 [cf. Exercise lB(a)], i.e. iff the homogeneous linear equation (2.3.3)

£/?(/) = 0

has a unique solution £, the zero linear combination 0 G 5 ( 5 , I i i e / 5 ) . In particular, the 0-tuple 0 -» V is independent. If (2.3.3) is underdetermined, i.e. / is not independent, then / is said to be dependent. If the unital ring 5 is non-zero, then if = jf for i # y e / implies that (2.3.3) has the non-zero

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solution £ = (1 -» It,-) - (1 -» liy). Thus a multiset (if\i e /> with any element of multiplicity more than 1 is necessarily dependent in this case. For arbitrary / : / -» V, the image Im R(f) of (2.3.2) is a submodule of V. This submodule is called the span of / or (if\i e />. Thus / spans in the sense of Definition 2.3.2(S) iff Im /?(/) = V. Example 2.33. Let 5 be a unital commutative ring, considered as a right S-module Ss. (a) The singleton {s} £ S is dependent iff s is zero or a zero divisor (cf. Exercise 1.4M). (b) The singleton {s) spans iff s is invertible. (c) The singleton {1} is a basis. (d) In Z, the subset (2-element multiset) {2,3} is dependent, and spans. The subsets {2} and {3} are each independent, while neither spans. D Example 23.4. The right Z-module Z 2 n a s singleton {1} spans, but is dependent. D

no

basis. In particular, the

The existence of the isomorphism L : V -» S(S, V); v -» (L(v) : s <-* vs) may be reformulated as a freeness property, analogous to (O 4.1.2), Exercise I 1U, or Section I \A. Given a set map 0 : {1} -» V; 1 >-* v, there is a unique 5-homomorphism B = L(v): S -* V with 1 0 = 1 0 . This property is expressed by saying that the right S-module S s is the free S-module on the generating set {1} (cf. Example 2.1). Now consider the index set I as the disjoint union T.¡ e ¡{j} of its singleton subsets {)}. The disjoint union of the composites {;'} -* {1} •-> 5 —'-* L t , e / 5 , with the insertions i¡ as in (1.1.5), defines an insertion i : / - » LI i e /5. Suppose a set map 0:I~*W to an S-module W is given. For j in /, define 6j : {1}_-» W;l-* jO. The freeness of S on {1} yields a unique S-homomorphism 9} :S -* W with 10, = 10-, i.e. with 10y =j&. There is then a unique S-homomorphism 0: UieIS -* W with JL6 =j0 for each j in /, namely 0 = Lie¡6¡. Indeed jid = li ; 0 = 10y = jd. Summarizing: there is an insertion i : / -» LI l 6 /S such that, given a set map 0 : / -» W, there is a unique S-homomorphism 0 : L1 1S/ S -» W with 40 = 0. One says that LI i 6 ¡S is a free S-module on the generating set I. (Cf. Example 2.4 for the case of finite I.) The property may be expressed diagrammatically as: ie,

r

w an analogue of (O 4.1.3).

,

w

VECTOR SPACES AND MODULES

147

Now suppose that an 5-module V has a basis / : / -» V, so that (2.3.2) yields an isomorphism (2.3.5)

US-^S(S,US)^*V

of 5-modules. Note that iLR(f) = f:I-*V. Given a set map 6:I-*W from / to an 5-module W, there is a unique 5-homomorphism 6 : LI íe ¡ 5 -* W with i0 = 8. Thus there is a unique 5-homomorphism 6 = (Li?(/)) -1 0 : K -» W with /Ô = iLR(f)d = L6 = 8. Diagrammatically, one has the analogue

i-L-v W— W of (2.3.4) and (O 4.1.3), so one is also justified in calling V a free S-module, with basis f: I -* V. In this sense, L J , e / 5 is a free 5-module with basis i : / -> LI / e / 5 (cf. Exercise 2.3G). If F is a free 5-module with basis f: I -* V, then the inverse R(f)_1 of the isomorphism /?(/) : 5,, -» F of (2.3.5) yields an isomorphism (2.3.7)

R(f)'1 :V^Sl;v~

For Í in /, the scalar HvRif)'1) respect to the basis f:I-* V.

(i -

i(vR(fy1)).

e 5 is called the i-th coefficient of v with

Example 2.3.5. For an /-linear combination £ in the free module 5¿ with basis t : I -* 50; i -> i¡, the i-th coefficient of Í is the element if of 5. D Example 2.3.4 shows that, for an arbitrary ring 5, an 5-module V need not possess a basis. In other words : not all modules are free. For a vector space V over a field K, however, the following theorem shows that V does have a basis. Thus all vector spaces are free. Theorem 23.6. Let V be a vector space over a field K. Then V has a basis. Proof. Let {A, ç ) be the poset of independent subsets of V, with order induced from the power set (&(V), c). The poset (A, ç ) is inductive, since the union U = (JC of a chain (C, ç) of independent sets is independent. Indeed, if ÇR(1V) = 0 with f = (T? : X -» K*) U((tf - X) -+ {0}) and finite, non-empty X, then X would be a subset of an element W of C. But then ((TJ : Z -> #*) U((JK - X) -► {0}))A(V) = 0 would contradict the independence of W. Now by Zorn's Lemma O 3.5.3, (A, ç ) has a maximal element B. As an element of A, the (multi-)set B is independent, i.e. R(1B) injects. It

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remains to be shown that R(1B) surjects. For b in B, one has (({6} -» {1}) U((B - {b}) -► {0}))/?(lB) = b. For v in V- B, one has B c B U {w} = 5 U{Ü}. Since 5 is maximal in A, the set B U {i>} cannot lie in A, and must thus be dependent. Consider a non-zero solution f = ( / - * K * ) u ((B - / ) -* {0}) U({u} -» {A:}) to f/?(l Bu{u) ) = 0, with / a finite subset of B. Since B is independent, k # 0. Then y = E ; e yjf • (-/'f • k'1) e Im fi(l;) ç Im i?(l B ). Thus /?(1B) surjects. D Corollary 2.3.7. A linearly independent subset I of a vector space V is a subset of a basis B of V. Proof. The element / of A is contained in a maximal element B of A, since Zorn's Lemma applies to the inductive subposet of A whose elements contain /. As in the proof of Theorem 2.3.6, the (multi-)set B is then a basis for V. U Determination of a basis B containing a given independent set / as a subset is known as extending the independent set / to the basis B. The process is used, for example, in proving the final theorem of the next section.

EXERCISES 2.3A. Let b : I -* / be a bijection of an index set /. Show that an /-tuple f'.I-* V has each of the properties of Definition 2.3.2 iff the /-tuple bf: I -* V has the corresponding property. 2.3B. Verify the statements of Example 2.3.3. 2.3C. Verify the statement of Example 2.3.4. 2.3D. For each element a of an index set A, let / „ : / „ - » Va be a basis for a right 5-module Va. Show that the disjoint union

aeA

V

ßeA

with ta as in (1.1.5), is a basis for \laeAVa. 2.3E. Let V be the real vector space R3 = Rj. An /-tuple / of elements of V is an element [f¡¡\ of V = R(R,R3)' s R(©¿.,R, ©f., R) s R3. An /-linear combination f is an element [ ^ j , . . . , ¿,] of U[ = R(R, ©¿„j R). Using e.g. Gaussian elimination to solve the homogeneous system [ £ , , . . . , £,][/;,] = [0,0,0] of 3 equations in / unknowns [viz. (2.3.3)], determine for each of the following / X 3-matrices [/j;],X3

VECTOR SPACES AND MODULES

149

whether / : { ! , . . . , / } -* Ul is linearly independent:

(«)

1 2 0

2 2 1

0 -2 ; 1

(*)

i 2 0

2 2 1

0 1 1

(0

1 2 0 2

2 2 1 0

0 1 1 1

2JF. For / ç /, let (g : / - V) = ( / : / -* F ) L (a) If g spans, prove that / does also. (b) If / is independent, prove that g is also. (Hint: Cf. Exercise O 3.1F.) 2.3G. For the /-tuple i : / - » L I i e / 5 , verify that /?(i): S(S, LI, !e/ 5) -> I i , e / 5 fan instance of (2.3.2)] bijects. Hence verify that i:I-* LI i e / 5 is a basis of LI i e ¡S. 2.3H. Let F be a vector space over a field K. Show that an /-tuple f: I -* V or multisubset (if \i G /> of V is dependent iff 3 i G /. if G Im /?(/!/_{,}), i.e. iff one element of the multiset lies in the span of its complement. 2JI. Let 5 be a unital ring. Show that for each (unital) 5-module V, there is an exact sequence O^R 2JJ.

2.3K. 2.3L. 2.3M.

0

of 5-modules with F free. Let S be a unital ring. A (unital) right 5-module V is projective if, for each surjective 5-homomorphism g : W -* X and h G S{V, X), there is an 5-homomorphism h:V -» W with Jig = h. [The homomorphism h G S(V, X) is said to have been lifted to the homomorphism 7i G S(V, W).] Show that free 5-modules are projective. Show that an 5-module V is projective iff it is a direct summand of a free module. For a field K, consider the ring 5 = K2. Show that the 5-submodule K = K X {0} of the 5-module 5 S is projective, but not free. Prove that an 5-module Q is projective iff each short exact sequence 0 ->A -* E -> Q -> 0

of 5-modules splits (cf. Exercise 1.1D). (a) The concept of an injective right 5-module V is dual to the 2.3N. concept of a projective right 5-module. Use this information to formulate an exact definition of what it means for an 5-module V to be injective. (b) Verify the validity of the definition given in answer to (a) by proving that an 5-module A is injective iff each short exact sequence 0-*A-*E-*Q-*0of 5-modules splits.

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2.30. Show that the abelian group (Q, +,0) of rational numbers is an injective Z-module. 2.4 Determinants and Dimension Throughout this section, let K be a commutative unital ring. If F is a tf-module, then the conjugate (or "dual") V': = K{V,K) or K(V\K) or K W; K) is again a .K-module (cf. Exercise 2.1D). Elements of V are known as functionals on V. (For example, Exercise 2.1 H exhibits the Heaviside function as a functional on the real vector space C(I) of continuous functions on the interval /.) Now let U and W be further (unital) /C-modules. Recall the Currying bijection (2.4.1)

{UW)V -*UVxW;(g:v*(fv:w~>u))~(h:(v,w)

- u)

of (O 3.4.1). The image of the subset K(V, K(W, U)) under this bijection is called the set K(V, W;U) of bilinear junctions from Vx W to U. Elements of K(V, W;K) are called bilinear pairings of V with W. Linearity of /„ in (2.4.1) corresponds to linearity of h in its second slot. Linearity of g in (2.4.1) corresponds to linearity of h in its first slot. The restriction of the bijection (2.4.1) to the subset K(V, K(W, U)) gives a bijection from K(V, K(W, U)) to K(V, W; U). This bijection transfers the A-module structure on K{V, K(W, 10) (cf. Exercise 2.1D) to K(V, W; U). In the case V = W and U = KK, the module K(V, V; K) is called the module K\V; K) of bilinear functionals on V. For example, the usual Euclidean inner product (x, v) -» xyT on U\ is a bilinear functional on R3. The signed area xux22 ~ *nx2i °f t n e parallelogram (0,0),(*,,, xn), (x21, x22), (xn + x21, xl2 + x22) spanned by an ordered pair (Cxn, *12),(*2i> x22)) of vectors in R2 yields a bilinear functional ((xn,xl2\(x2l,x22))'-+xux22-xi2x2l on U2. The signed area vanishes when the two vectors are equal. In general, a bilinear functional on a /C-module V is said to alternate if its restriction to the diagonal V in V1 is zero, i.e. if it vanishes on the complement of the binary diversity relation on V. The set K2(V; K) of 2-forms or alternating bilinear functionals on V is a submodule of K2(V; K). For a natural number n, consider the bijection {{...{KV)...YY -* vxvx-xK pj f y instance of the extension of (O 3.4.1) or K w ¡ t n n CO es 0 t an (O 3.4.3) as in Exercise O 3.4D. The image of K{V, K(V,..., K(V, K)...)) under the bijection is the module Kn(V; K) of n-linear functionals on V. Elements of the submodule K"(V; K) of n-linear functionals vanishing on the complement of the n-ary diversity relation on V, i.e. vanishing whenever at least two of their arguments agree, are known as n-forms or alternating «-linear functionals. Note that the semigroup Set(Z„,Zn) and symmetric group Sn have left actions on K"(V; K) given by (2.4.2)

T:h~(7h:(Vlt...,VH)~>h(V„,...,VHT))

VECTOR SPACES AND MODULES

151

for T : Zn -> Z„, with K"(V; K) as an invariant subset. If h is an n-form and T is the transposition (/;), then 0 = /Í(Ü, v¡ + vjt...,v¡ + v¡,...,vn) = h(vx v¡, . . . , Vj vn) + h(vx, . . . , Vj, . . . , v¡, . . . , vn) = h(uu ...,v¡,...,v¡ vn) + Th (u„ . . . , v¡,.... Vj,..., v„) implies Th = (-l)A. Since K"(V; K) is an 5n-set, it follows more generally that Th = h for even T and Th = -h for odd T in S„. Thus for /i in KniV; K) and T e Set(Zn,Z„), one has (2.4.3)

Th = if T € 5„ then (sgn T)/J else 0.

In a different direction, elements of ¥ix(K"(V; K), Sn) are described as symmetric n-linear functionals. A Ä-homomorphism 6: V-» if induces a #-homomorphism E " _ l 0 : V -* Wn. One then obtains a £-homomorphism (2.4.4)

^ ( í ; * ) : * " ^ ; * ) - K"(V; K); h ~ I ¿ ô) A.

In other words, £"(0; K)A(i>i vn) = M ^ ö , . . . , y„ö) for (u 1 ; ..., vn) e K". If A is an n-form on W, and v¡ = uy, then tfn(0;K)h(...,u¡,...,v¡,...) H = h(... ,v¡6,..., Vj6,... ) = 0. Thus K (6; K) restricts to a /C-homomorphism Kn(6; K) = K"(W; K) - Kn(V; K). If

V is also a #-homomorphism, then / " M ; K) = ¿"(0; K)Kn(
t fiJmE\!¡,...,

Wi-i

= E A-i

/»-i

t

i.-i

Ljßli] I

ÍfiJl-f»,kdet(EÜ,...,E¡í-) fulT...fn,nTTkd*(E\l...,E\:)

E 7,e§eL(Z„.Z„)

=

E fl.XT--fn.nAm

=* £

T)k dCt(7)

(sgnr)/liir.../„nr,

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CHAPTER II LINEAR ALEGEBRA

and is thus specified uniquely. On the other hand, defining (2.4.5)

|/|=

Ylfiju

E (sgnt/) UeS„

'e£n

does give an n-linear functional on K", taking the value 1 on /. Now suppose that the i-th and ;'-th rows of / agree, for Í # j . Then 1/1= = =

E (sgn U) f i fuu

UeS„

'eZn

E (sgn U) El fk.kv +

E

O*n£0n/*.H/

(sSflT)flilT...fUT...fjjT...fnnT

E

TBA„

- E (sgnr)/ 1 - i r ...y;- i ; T .../ > > i r .../ W i „ 7 .-o re/l„

since /) t = fjk foi 1 <, k £ n. Thus defining k det : ii£ -* K; f -* k\f\, one obtains the unique element of Kn(K"; K) with k det / = /:, as required. D Definition 2.4.2. For an « X n-matrix / e K'¿, the scalar det / or | / | given by (2.4.5) is called the determinant of /. D Theorem 2.4.3 (Properties of the Determinant). Forf, g e K',¡:

(a) l/gl = l/l-lgl; (b) |/rl = l/l. Proof, (a) For /?(g): K[' -> K['; v -» i#, one has K"(R(g>, /Odet I = det //Kg) = det g. Thus K"(R(g);K) det = det g det by Proposition 2.4.1, whence det fg = Kn(R(g); /Odet / = det g • det /. (b) By (2.4.5), one has l/rl=

E (sgn £0 n

UeS„

<eZ„

flu

= E (sgnt/) n/ it /., =

E i;-'ei„

( s g n i r ' j sn2 / U r . - l / l -

°

J' »

Theorem 2.4.4. For eacn natural number n, the module Kn(K"; K) ofn-forms on the free module K" is the free module with singleton basis {det}.

VECTOR SPACES AND MODULES

153

Proof. For h G K"(K"; K), one has h = A(/)det by Proposition 2.4.1. D Proposition 2.4.5. Ifr> n, then Kr{Kn; K) = 0. Proof. For h G Hr(Kn; K) and / G K?, one has

fh=h\ ÍflhEÜ,..., Wl-l

ifrjEljA /r-l

/

= E -

Íflh~-fr¡HElJ»l>->Eü)

=

h.iT---fr.rrh{E\^

E

7-e§gL(Zr,Z„)

£ 1 V T ) = °.

since r > n implies V J G Set(Z,,Z„), 31 < i < ;"^ r. iT = jT.

D

Theorem 2.4.6. Let K be a non-zeroring.Let Vbe afreeK-module with a finite basis ß:Zn -* V. Then the cardinality n of the basis is uniquely determined by V. Proof. By (2.3.2), there is a ^-isomorphism R( ß ) : K" -* V. Suppose that V also had a basis y : Z, -» V with r > n, yielding a /^-isomorphism R(y):Kr -* V. Then R(y)R(ß)-1R(ß)R(y)-i = 1*,, whence the identity map Kr(\Kr\ K) on Kr(Kr; K) would factor as (Kr(R(ß)R(y)-u, K): Kr(Kr; K) -> K'(Kn; K)XKr(R(y)R( ß)~l; K) : Kr(Kn; K) -> Kr{Kr; K)). But this is impossible, since Kr(Kn; K) = 0 by Proposition 2.4.5, while Kr(Kr; K) = KK*0by Theorem 2.4.4 and the hypothesis. D Definition 2.4.7. If a free /C-module V has a basis ß : Zn -» V, then F is said to have rank or dimension dim V = n. D Consider an endomorphism 9 : V -» V of a free module with bases /3:Z„->K and y : Z„ - K Note that /?(?)/?( 0 ) - ' . R(ß)R(y)-> = Ä C y H ^ y ) - 1 = I*». Identifying K(Kn,Kn) s ÀT„", Theorem 2.4.3(a) yields \R(.y)0R(y)-l\ = |Ä(y)K( /3)" » • Riß)dR(ß)-1 ■ R(ß)R(y)-i\ = 1 1 1 \R(y)R(ß)- \ ■ \R(ß)6R(ß)- \ ■ \R(ß)R(yr \ = \R(y )R( ß )~ '| • I M ^ i y r ' H Ä i ^ e M ß ) - 1 ^ |Ä(/3)ÖÄ(/3)-'|. One may thus define the determinant of the endomorphism 6 unambiguously by det 6 = \6\ ~ \R(ß)6R(ß)~1\, independently of the choice of basis ß : / -» V. In particular, d e t l K = 1. Moreover, for a second endomorphism 3)-1| • l \R(ß)(pR(ß)~ \ = IÖ| • |
J

A

AT", define a new matrix / = [/,,]„><„ by

(2.4.6)

/;=

E

sgni/ n

/*.«,.

J

*} "

A

"

154

CHAPTER II

LINEAR ALEGEBRA

Now the (i, O-entry of the product matrix / / is n

n

;-l

7-1

E/yZ/i = Lfij =

E

Ue(in)SH.t(.nj)

sgni/ n

fh.hu

i*AeZ„

E sgn £7 n / i . i i / = 1 / 1 .

since S„ = Uyez^" 1 ^«-^"/)- F° r ■ '' * ^> l e t S* De t n e matrix obtained from / by replacing f¡¡ with / t ; . Then, as above, one has E"_i/fc;^f = 1**1 = tí I = 0- I n (2.4.7)

other words

' Z'-Jkjfji

= sik- \n or

ZZ = I Z K

in K;;. Proposition 2.4.8. For f in K"t, the following conditions are equivalent: (a) 3*. fg=l; (b) 3g.gf=U (c) / is invertible, (d) l/l is invertible. The conditions are also equivalent when applied to endomorphisms of a free K-module of finite dimension. Proof. If (a) or (b) hold, or (c) holds such that / has inverse g, then \fg\ = \fl\g\ = \gf\ = 1, so (d) holds. Conversely, if (d) holds for a matrix^/, then (2.4.7) shows_that / ( l / l - 1 / ) =ff- IZI"1 = / and (\f\~1fTT)f= \f\~xfTT ■ Z r r = IZr 1 (Z r -Z T ) = I Z r 1 - I Z r l - / = /. so that / is invertible. Finally, if ß : Z„ -» V is a basis for a 7?-module V, then the various conditions on endomorphisms 6 translate to the corresponding conditions on the matrices R(ß)eR(ß)~i. D If K is a field, then part (c) of Theorem 2.4.9 below gives an alternative approach to the equivalence of (a) and (b) in Proposition 2.4.8 above. Theorem 2.4.9. Let K be a field, and let 8 : V -» W be a K-homomorphism. (a) If V has finite dimension, then dim V = dim Ker 8 + dim Im 8. (b) / / W has finite dimension, then dim W = dim Coker 8 + dim Im 8. (c) If dim V - dim W is finite, then 8 injects iff it surjects. Proof, (a) Pick a basis K: J -* Ker 8 for Ker 8, and extend to a basis (K U A) : / U L -> V of V. Then À0 : L -* V8 is a basis for Im 8. Indeed,

VECTOR SPACES AND MODULES

155

for v € V, one has u8 = (UR(K U A)_1X(0: U y 5 -» LIy 5) + (1 : LIj. 5 -> UL S))R(\6\ so A0 spans Im 0. On the other hand, for r¡ e S(S, UL 5) £ S(S, U y u L 5 ) , the equation TJ/?(A0) = 0 implies 7jÄ(A) e Ker0, i.e. 3£ e 5(5, LI, 5) <. S(S, UJUiLS). CRU) = 77ÄU), whence (£ - TJ)Ä (K U A) = 0. Since K U A is a basis for F, it follows that £ - 77 = 0 or £ = 77 e (Lly 5) n (LIL 5) = {0}, i.e. 77 = 0. Thus A0 is independent, and so a basis for Im 0. Then dim F = \J u L| = | / | + |L| = dim Ker0 + dim Im0. (b) The proof of (b) is dual to (a). Pick a basis K : J -* Im 0 of Im 0, and extend to a basis ( K U À ) : / U L - » ^ of W. Then the composite of A : L -> W with the natural projection W -* Coker 0 ; w - » w + l m 0 i s a basis for Coker 0 (Exercise 2.40). (c) If 0 injects, then dim Ker 0 = 0. Thus dim Im 0 = dim V = dim W. It follows that Im 0 = W, for otherwise a basis for Im 0 would have a proper extension to a basis for W, violating the equality of the dimensions. Thus 0 surjects. In the other direction, suppose that 0 surjects. Then dim Ker 0 = dim V — dim Im 0 = 0, so that 0 injects. □ EXERCISES 2.4A. For a ÜC-module X, show that the evaluation map of Exercise 2.IE, with G = K, furnishes a bilinear pairing of X with its conjugate X' = # ( * , K). 2AB. (a) Let / be a fixed element of K.%,. Show that matrix multiplication gives a bilinear pairing (f, 17) -» Çfri of /C,m with Kj,. (b) Show that / -» (( f, T/) -» f/77) gives a iC-isomorphism between A"£ and UK?, K\\ K). 2.4C. Show that the Riemann integral C(I) X C(I) -* R; (/, g) -» f,f(x)g(x)dx gives a bilinear functional on the R-module C(/) of Example 2.1.5. 2.4D. For a fixed 2-linear combination ¿¡eK2, show that the function V2 -► F; / -» £/ [as in (2.3.1)] is bilinear. 2.4E. Verify that (2.4.2) makes A-"(F; AT) a left Set(Z„,Z„)-set and a left 5„-set. Verify that Kn(V;K) is an invariant subset of each. 2.4F. Express each bilinear functional on a real vector space as the sum of a symmetric bilinear functional and an alternating bilinear functional. 2.4G. (a) Show that the determinant of a diagonal matrix is the product of its diagonal entries. (b) Show that the determinant of a shear matrix /„ + kE'nJn (i # j , and Jt e K) is 1. 2.4H. For T e 5 n , let pT be the corresponding permutation matrix as in Exercise 1.21, with A = K. Prove that det pT = sgn T. 2.41. For what values of n is the determinant of an n X n-matrix / e K" given by Sarrus' Rule det f=LTeDn (sgn T) n , s z / /t ,T?

156

CHAPTER II

2.4J.

Give an example of a unital, non-commutative ring 5 and distinct natural numbers m, n such that the 5-modules Sm and S" are isomorphic. Let GL„(K), the general linear group of dimension n over K, be the group of invertible elements of the monoid (Jf£, •, /„). Show that matrices in the same (group) conjugacy class have the same determinant. Let SL„(/0, the special linear group of dimension n over K, be the set of elements of Gh„(K) whose determinant is 1K. Show that SL n (iO is a normal subgroup of GLn(K). What is the quotient group GL n (tf)/SL n (/0? Exhibit a group isomorphism SL2(Z2) = S3. Let 0 -* A0 -* Ax -* ••• ->i4„ -»0 be an exact sequence of finite dimensional vector spaces. Prove that E"_ 0 (- D'dim A¡ = 0. (Cf. Exercise IF.) Complete the proof of Theorem 2.4.9(b). For vector spaces U and V over a field K, prove that U < V and dim U = dim V < « implies U = V. Let C denote the subset

2.4K.

2.4L.

2.4M. 2.4N. 2.40. 2.4P. 2.4Q.

LINEAR ALEGEBRA

x -y

y x,y x

e

of R\. Elements of C are known as complex numbers. (a) Show that C is a commutative subring of U\, containing the ring of scalar matrices as a subring isomorphic to, and identified with, the ring of real numbers R. (b) Set i = 1 ¿1. Show that i' 2 = - 1 . (c) Show that each complex number z can be expressed uniquely in the form z = x + iy, with real part x = \ tr z (cf. Exercise 1.3E) and imaginary part y = -i(z - ^tr z). (d) The transpose zT of a complex number is called its complex conjugate z. Show that transposition or conjugation yields an automorphism of the ring C. (e) Prove det z = zz and tr z = z + z. (f) Prove that the determinant of a complex number z is non-negative. The absolute value or norm or modulus of z is Vdet z. It is denoted by \z\ (not to be confused with the notation |z| for det z). Prove \zw\ = |z|-M(g) Show that C is a field.

VECTOR SPACES AND MODULES

157

2.4R. Let m and n be positive integers that can be expressed as the sum of two squares of positive integers, e.g. 10 = 3 2 + l 2 and 20 = 42 + 2 2 . Prove that the product mn can be expressed similarly. [Hint: Use Exercise 2.4Q(f).] 2.4S. Let D-0 denote the subset t X

y

z

X

t z -y

y -z t X

z y ~x t

t,x,y,z

e IR

of U*. Elements of H are known as quaternions. (a) Show that H is a subring of R*, containing the ring of scalar matrices as a subring isomorphic to, and identified with, the ring of real numbers IR. (b) Set

1 =

0 -1 0 0

1 0 0 0

0 0 0 1

k=

0 0 0 1

0 0 1 0

0 -1 0 0

0" 0 -1 . 0

;-

0 0 1 0

0 0 0 -1

1 0 0 0

0 1 0 0

Show that / = j = k = — 1, ij = k, jk = i, ki =j, ji = —k, kj = —1, and ik = —j. (Compare Exercise I 2.5E.) (c) The transpose qT of a quaternion q is called its conjugate q. Show that transposition yields an isomorphism of H with H"9. (d) Prove that HI is a skewfield. 2.4T. (Loop Transversal Codes over Finite Ring Alphabets.) Let A be a commutative unital finite ring, regarded as an alphabet. Let T be a IA\n~k-element subset of the free y4-module A" such that T contains 0 and a basis B for A". Suppose that T carries a unital A -module structure (T, *, A) given by an ^-isomorphism s : (T, *, A) -* (An, +,A) such that s :(T, +, A) -> (An, +, A) is a partialA-homomorphism. In other words, for t, u G T and a e A, one has t + u e T =» (r + u)s = ts + us and toeN (ta)s = tsa. Show that there is a linear code C, a free submodule of dimension k in A", to which (r, *, 0) is a loop transversal. Moreover, show that T is precisely the set of errors corrected by C. (Hint: Generalize Theorem I 4.4.4.)

158

CHAPTER II

LINEAR ALEGEBRA

2.4U. Express f¡¡ from (2.4.6) in terms of the determinant of the (n - 1) x (n — l)-matrix obtained from / by removing its i-th row and )-th column. 3. COMMUTATIVE ALGEBRA A module over a ring is the linear-algebraic analogue of a set with a monoid action. Linear algebraic analogues of monoids are known as "algebras." (The rather grandiose name probably arose because early twentieth-century algebra was almost exclusively devoted to the study of "algebras" in this sense.) If K is a unital commutative ring, then a K-algebra (AK, •, 1) is a unital K-module AK and a monoid (A,-, 1) such that the monoid multiplication A X A -> A; (a, b) -* ab is bilinear. In other words, the right regular representation R : A -» K(A, A); b -> (x -» xb) and the map L : A -* K(A, A); a -» (x -* ax) are K-homomorphisms. Note that a Z-algebra is just a unital ring. Note that K itself is a K-algebra (KK, •, 1). The following proposition describes a copious source of ^-algebras. Proposition 3.1. Let f:K-+Sbea unitalring(£, •, 1) such that (3.1)

unital ring homomorphism from K to a

Vk e i , V s e 5, kfs = skf.

Then the composite of f : K -* S with the right regular representation R: S -* End S of S makes S arightK-module SK and a K-algebra (SK, •, 1). Proof For s in S, one has R(s) e K(SK, SK), since Vx e 5, V i e K, (xk)R(s) =xkfs=xs-kf = (xR(s))k by (3.1). Then R : S -* Ar(5x, 5 K ) is a X-homomorphism, since Vx e S, Vfc e K, xi?(s&) = JC • skf = xs • kf = (xR(s))A: = x(Ä(s)i). Similarly L(i) G ATÍS^.Í^-), since Vx G S, Vk e= K, (xk)L(s) = s-xkf = sx-kf = (xL(s))k. Finally, L:S -* K(SK,SK) is a Khomomorphism, since Vx G 5, VA: e K, xL(sk) = skf • x = sx • kf = (xL($))Jt = x(L(s)*) by (3.1). D Example 3.2. For a commutative unital ring K and a positive integer n, the unital ring homomorphism f:K-* K" embedding K as the ring of scalar matrices makes K" into a K-algebra according to Proposition 3.1. In this context, one speaks of the matrix algebra K". D Example 33. For an interval / in the real line U, let C(/) be the set of continuous functions / -* R. Note that C(/) is a commutative unital ring under pointwise operations (cf. Exercise 1.4Q). Let / : 1R -» C(7); r -+ (x -» r) embed IR as the ring of constant functions. Then Proposition 3.1 makes CO) into an R-algebra. D

COMMUTATIVE ALGEBRA

159

Example 3.4. The complex numbers C (Exercise 2.4Q) and the quaternions H (Exercise 2.4S) form IR-algebras according to Proposition 3.1, taking the homomorphism / in each case that embeds R as the ring of scalar matrices. D

Given two ^-algebras iAK, •, 1) and (BK, •, 1), a K-algebra homomorphism h : (AK, •, 1) -» (BK, •, 1) is a function h: A -» B that is a iC-homomorphism h: AK -» BK and a monoid homomorphism h : (A, ■, 1) -* (B, ■, 1). For a given unital commutative ring K, the most fundamental ^-algebra is the polynomial ring K[T] in the indeterminate T. Let TN be the free monoid on the singleton T (cf. Exercise O 4.2C). Then the ¿[-module K[T]K is the coproduct LI TnK, with insertion ir» : K -* /¡f[r]; A: -» kT". In particular, i, embeds K in /¡T[T]. Since 7** is a basis for ^[T]^, each polynomial piT), i.e. element of K[T], has a unique expression pit) = pnT" + ■■• +/?,r+jp 0 , with pn # 0 unless / J ( D = 0. For non-zero piT), the natural number « is called the degree deg piT) of p(T), and p„ is called the leading coefficient of piT). If n > 0 and pn = 1, then piT) is said to be monk. Further, degO : = -oo. Note degipiT) + qiT)) <. max{deg piT),deg qiT)}. The element \iT* = T" of KiTn is called the monomial of degree n. Elements of K = Ki1 are called constants. Note that the degree of a monomial Tn is just its length as an element T" of the free monoid TN. Now the right regular representation RiT") : r N -> TN of an element T" of TN concatenates with the embedding r N «-» #[7"] of monic monomials to yield a set mapping RiT"):TN -> A"[r]. Since AT[r] is the free ^-module on TN, there is then a unique Ä-homomorphism RiT"): K[T]-» K[T] restricting to RiT") : TN -* K[T] on TN. Finally, the set map R:TN - # ( # [ 7 1 * , #[71*.); T" •-> RiT") extends to a unique K-homomorphism R:K[T]->K(K[T]K,K[T]K),

(3.2)

the ng/if regular representation of #[7"]. Proposition 3.5. The multiplication piT) • qiT) ■■= piT)RiqiT)) makes the polynomial ring K[T] into a commutative K-algebra iK[T]K, ■, 1). Proof. For natural numbers m and n, both RiTm+") and RiTm)RiT") in KiK[T], K[T]) restrict to the same map TN -* AT[V]. Thus Ä : 7"N -+ AT(.K[7,]jC,.flf[r]Jf) is a monoid homomorphism, with commutative image. Now for piT) and qiT) in ir[r], one has

piT)qiT)=piT)RiqiT))

= 9(nMn.

/ deg p

=\ E ^ \ i-o

\

/deg q

W E i/

I

\ /-o

J

\

deg p deg q

- E /

L Pi
i-o y-o

160

CHAPTER II

LINEAR ALEGEBRA

so that K[T] has a commutative, bilinear multiplication. Also

(

=

deg p deg q

\

E £/w,r'+>

deg p deg ? i-O j - 0

/

E EMJAÍ^')

i-0 / - 0

= F Z P*iR{Tl)R{Ti) /-o y-o

' deg p

=

E A*(n

'deg q

Zq¡R{T>)

. '-0

=

R(p(T))R(q(T)),

so the multiplication in K[T] is associative. D Make {-<»} U N a semigroup under + with {-<»} as a sink. Then deg( p(T)q(T)) ^ deg p(T) + deg q(T), with equality guaranteed if either of p(T), q(T) is zero, or otherwise if neither of the leading coefficients of p(T) or q(T) is a zero divisor. In particular, equality always holds if K is an integral domain, so that K[T] is then itself an integral domain. Furthermore, (3.3)

K[T]*=K*

in this case, since p(T)q(T) = 1 => 0 ^ deg p(T) + deg q(J) = deg p(T)q(T) = deg 1 = 0, whence deg p(T) = deg q(T) = 0 and p0q0 = 1. The importance of the polynomial algebra K[T] rests on the following universality property: Given an element t of a ^-algebra S, there is a unique Ä-algebra homomorphism (3.4)

tfin-SjpíT)-/^)

mapping T to i. Indeed, for p(J) = E ^ ' p , ! ' ' , one has pit) = E£g„'P¿ f'A first application of the universality property takes the ¿-algebra S to be the set Set(K, K) of set functions from K to K, with the pointwise unital ring operations of unit 1 : x •-» 1, subtraction g - h: x ^ xg - xh, and product g • A : x ~ xgxh. Let / : K -* Seti/sT, X); * ~ (* -» £) embed K as the subring of constant functions. By Proposition 3.1, Set(iT, ÜT) becomes a K-algebra. For the element \K : x •-» or of Set(^T, /O, the universality property (3.4) gives a unique ÍC-algebra homomorphism (3.5)

K[T]^Set(K,K);p(T)~(k~p(k)).

The image of a polynomial p ( D under (3.5) is described as the polynomial function p(x) : K -» K; x -> /?(*). Elements of the preimage {* e £ | p ( x ) = 0}

COMMUTATIVE ALGEBRA

161

of 0 under pix) are called roots of the polynomial p(T) in K[T]. Note that distinct polynomials in K[T] may yield the same polynomial function. For example, T3 + 27 is a non-zero polynomial in Z3[T] yielding the zero polynomial function Z 3 -» Z 3 . A useful tool for designing polynomial functions over fields is provided by Lagrange interpolation. Proposition 3.6. Let Kbe a field. (a) (Lagrange Interpolation.) Let X = {jCj,..., xn) be a finite, non-empty subset of K. For 1 <, i £ n, define the delta polynomial

(3.6)

w-n—T-

Let f: X -* K be a function. Then the Lagrange interpolant polynomial T.".if(Xi)ôxfT) yields a polynomial function extending f. (b) If K is finite, then each element of SetjK, K) may be realized as a polynomial function. Proof, (a) The delta polynomial (3.6) yields a polynomial function K -* K restricting to the delta function X -» K; x <-* if x = x¡ then 1 else 0 on X. The value of the Lagrange interpolant polynomial function at x¡ in X is then E"_,/(jr;)Sx(jc,) = /(*,), agreeing with the value of / there. (b) Take'* in (a) to be K. D Since the polynomial ring construction applies to arbitrary commutative unital rings, it applies to polynomial rings themselves. In particular, given a commutative unital ring K and indeterminates T, U, one may form the polynomial AT[r]-algebra K[T\U\. Consider a commutative X-algebra S and an element (/, u) of S2. By (3.4) for K[T], there is a unique ^-algebra homomorphism / : K[T] -» S; p(T) -» pit) mapping T to t. By Proposition 3.1, 5 is a K[T]-algcbra. By (3.4) for K[TIU], there is then a unique AT[r]-algebra homomorphism K[T]LU] -» S; ZjP¡(J)U' -» LjpjiTYu' = Lj(Lf¡*ip'Pj¡t')u' mapping U to u and restricting to / on K[T], i.e. mapping (r, U) to (f, u). Continuing by induction, one obtains the following: Theorem 3.7. Let K be a commutative, unital ring and let n be a natural number. Then there is a commutative K-algebra K[TX,..., Tn] containing a subset {Tlt..., Tn] such that, given a commutative K-algebra S and an element (¿i tn) ofS", there is a unique K-algebra homomorphism (3.7)

K[Tx,...,Tn]^S;p(T,

mapping each T¡ to t¡.

Tn)

~p(tlt...,/„)

□

Consider Set(K", K) as a unital commutative ring with pointwise operations. Let / : K -> SetiK", K) embed K as the subring {Kn -> K; (xv...,xn) <-* k\k e K) of constant functions. Proposition 3.1 makes

162

CHAPTER II

LINEAR ALEGEBRA

Set(K", K) into a commutative tf-algebra. For the element ((jc l f ..., xn) -» xJ7- • •, (*i> • • • 7 x„) -» *„) of Set(K", ^O". the universality property of Theorem 3.7 gives a unique ^-algebra homomorphism K[Tl,...,Tn]^Sçl(Kn,K);

(3.8) /»(^i

rw) -» ( ( * ,

Ac,,) -»/»(*„...,*„)).

the multivariate analogue of (3.5). EXERCISES 3A. Let A be a tf-algebra. (a) Show that f : K -* A; k ^ Ik is a ring homomorphism. (b) Show that Vite A:, Va eA,kfa (c) (d) (e) 3B. (a)

(b) 3C. (a)

= akf

[cf. (3.1)1 If # is a field and A # {0}, show that / injects. If K = Z and /I is a unital ring, show that there is a non-negative integer n, the characteristic of ^4, such that K e r / = nZ. Show that the characteristic of a field is zero or prime. Let A be a tf-module. Let M be the subset of K(A,A;A) consisting of those bilinear maps m : A X A -> A for which (/l K ,m,l) is a X-algebra. Show that M is a submodule of K(A,A;A). If A is a vector space of dimension n over a field K, determine dimKiA, A; A) and dim M. Show that f:C

-» H;

t —x

t X —x >-» t 0 0

X

t 0 0

0 0 t

0" 0 -x

X

t

is a unital ring homomorphism. (b) Verify the need for hypothesis (3.1) in Proposition 3.1 by showing that the method of Proposition 3.1 applied to the ring homomorphism / : C -» H does not make H a C-algebra. 3D. Polly and Queeny are waitresses. In a typical week, Polly takes r cents in tips with probability pr, while Queeny takes Í cents with probability qs. Consider the polynomials p(T) = Lp¡ T' and q(T) = Zq¡T> in R[T]. Show that the probability that the two waitresses' total combined tips in the week are t cents is given by the coefficient of T' in the product polynomial p(T)q(T).

COMMUTATIVE ALGEBRA

3E. 3F. 3G.

3H.

31. 3J.

3K.

3L. 3M.

3N. 30.

163

Give a direct verification of the associative law p(T)q(T) ■ r(T) = p(T) ■ q(T)r(T) in the polynomial ring K[T]. Verify the universality property (3.4). Show that the universality property (3.4) of K[T] is equivalent to the following freeness property : Given a set map 0 : {T}_ -» A to a Ä-algebra A, there is a unique X-algebra homomorphism 0 : K[T] -» A with TO = tO. (Remark: This interpretation shows that K[T] is the linearalgebraic analogue of the free monoid TN on the singleton {T), as in Exercise O 4.2C.) Show that K[T] has the following universal property: Given a unital ring homomorphism f:K-*S, and an element t of the (not necessarily commutative) ring S such that kf • t = t • kf for all k in K, there is a unique ring homomorphism K[T] -» 5; p(T) -* pf(t) mapping T to t and restricting to / on the subring K of K[T]. Give an example of a cubic (i.e. degree 3) polynomial in Z3[T] yielding a non-zero homomorphism of the abelian group (Z3, - , 0). (Simpson's Rule.) Let / : [ — 1,1] -» R be a continuous, real-valued function on the interval [-1,1]. Simpson's Rule approximates the Riemann integral f[-i,i]f(x)dx as the integral of the Lagrange interpolant polynomial function determined by the function {-1,0,1} - R; - 1 ~ / ( - l ) , 0 ~/(0),l ~ / ( l ) . Find the quadratic interpolant, and evaluate its Riemann integral over [ -1,1]. (Permutation polynomials.) Let K be a finite field. A polynomial in K[T] is a permutation polynomial if its polynomial function is a permutation of K. (a) Show that a polynomial of degree 1 is a permutation polynomial. (b) Show that the symmetric group 5 3 is the set of degree 1 polynomial functions over the field Z 3 . (c) Show that the degree 1 polynomial functions over the field Z 5 generate a proper subgroup of the symmetric group 5 5 . (d) Show that the degree 1 polynomial functions together with the degree 3 permutation polynomial functions generate the symmetric group 5 5 . (e) Give an example of a cubic polynomial over the field Z 5 which is not a permutation polynomial. Write out a careful proof of Theorem 3.7. Theorem 3.7builds K[TX,...,Tn]inductively as K[TX\T2]...[T„]. Give an alternative construction of the .K-algebra K[Tv...,Tn] using the coproduct LI X..K of K over the free commutative monoid on the set X = {Tj,..., Tn) of indeterminates. (Hint: Generalize Proposition 3.5.) Let U be a subset of R". Prove that {pi^ Tn)\V(M1(...,«„) G {/", p(uu..., u„) # 0} is a submonoid of the monoid (K[Tit..., Tn], •, 1). If K is an integral domain, prove that K[Tit...,Tn] is an integral domain.

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3.1. Fractions The field Q of fractions is probably the most sophisticated algebraic construct common to the experience of every educated person. Identifying 51/17 with 531/177 is no mean feat, and many students succumb to the temptation of linearity by believing that the sum of f and 7 should be (3 + 4)/(5 + 7). Some perverse consolation may be derived from the realization that one is dealing with the universal Z-algebra Q in which the elements of the submonoid Z - {0} of Z have been made invertible. This section studies a general version of the construction of fractions. Throughout, K is a commutative unital ring. Let M be a submonoid of a commutative K-algebra A. Consider the direct product monoid M xA. Define a relation a on M X A by (3.1.1)

(m1,ai)a(m2,a2)

** 3m G M. mxma2 = axmm2.

Then a is a monoid congruence on M x A. It is reflexive since M is non-empty, and symmetric since A is commutative. Suppose (mx, ax)a(m2, a2)a(m3, a3), say mxma2 = axmm2 and m2na3 = a2nm3 with m, n in M. Then mx(mm2n)a3 = mxm(m2na3) = mxm(a2nm3) = {mxma2)nm3 = (axmm2)nm3 = ax(mm2n)m3, so that (m,, a^aim^ a3) and a is transitive. Finally, the commutativity of A readily yields that a is a submonoid of (M X A)2. The quotient monoid (M x A)" is denoted by M~lA, and an equivalence class (m,a)a is denoted by m~la or a/m or %. Note the "cancellation" an/mn = a/m for n in M. Now M X A is a K-set under the action MxAxK-*MxK; (m, a, k) -» (m, aA:), and a is a K-congruence, so A/"l4 becomes a K-set with action A/~l4 X K -* M~XA; (a/m,k) -* ak/m. The well-defined subtraction mx

m2

mlm2

then makes A/ -1 A into a JC-algebra (cf. Exercise 3.1 A). The ^-algebra M~lA has the universality property given by the following: Proposition 3.1.1. There is a K-algebra homomorphism (3.1.3)

■q:A-*M-1A;a~>a/l

such that, given a K-algebra homomorphism f : A -* B with Mf <.B*, there is a unique K-algebra homomorphism (3.1.4)

f:M-lA^B;m-xa~>(mfyl

af

with -qf = /. Furthermore, Ker 17 = {a G A\3m e M. ma = 0} ami A/17 ^ (A/-U)*.

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Proof. Verification that / is well-defined and has the claimed properties is left as Exercise 3.1B. Now Ker 17 = T?_1{0} = [a G/1|0/1 = a/1} = {a eA\(l,0)a(X,a)) = {aeA\3meM. \ma = 0ml = 0}. Finally, for m in M, one has ( m / l X l / m ) = 1/1 = 1. □ Example 3.1.2 (The nationals). If M is the submonoid Z - {0} of the Zalgebra Z, then (Z - {0})_,Z is the field O = {m _1 n|n G Z, m G Z - {0}}. D

Example 3.1.3 (The Field of Fractions of an Integral Domain). Generalizing Example 3.1.2, let S be an integral domain (and in particular a commutative Z-algebra). Then (S - {0})-1S is a field, since each non-zero element a/b has an inverse b/a. By Proposition 3.1.1, KeKrç : 5 -> (5 - (0})-15) = {0}, so Tj embeds 5 in (5 - {0})_1S. The field (S - {0})"'5 extending S is called the field of fractions of the integral domain S. D Example 3.1.4 (Rationals over an Integral Domain). For an integral domain K, the polynomial algebra K[T] is an integral domain. The field of fractions of K[T] has the form (3.1.5)

K(T) = {p(T)/q(T)\q(T)*0}.

Its elements are known as rationals, or more colloquially as "rational functions" (cf. the usage of Example 3.1.6 and Exercises 3.1F and 3.1K). D Example 3.1.5 (Division by Zero). If M contains 0, then Ker TJ = A by Proposition 3.1.1. D Example 3.1.6 (Localization). Let P be a prime ideal of a commutative unital ring A. By (1.4.2), the complement A — P of P is a submonoid of A. The ring (A — PYXA is called the localization of A at P. For instance, consider the polynomial ring R[T] identified via (3.5) with a subring of the ring C(R) of Example 3.3. For a real number x, the set Px = {p(T)\p(x) = 0} is a prime (indeed maximal) ideal of U[T], namely the kernel of the ring homomorphism R[7] -> R; p(T) -> p(x) of (3.4) onto the field R. Then the localization (U[T] - Px)~lU[T] comprises those rational functions that are well-defined in a neighborhood of the point x, i.e. "locally" around x (cf. Exercise 3.IF). D It is instructive to study the invertible elements of the result (A - P)~XA of the localization process of Example 3.1.6 applied to a prime ideal P of the commutative, unital ring A. For m " ' u e U - P)~lA with u € P, one has {m-luXu~lm) = 1, so m~lu G {A - P)~\A - P) =» m~xu G {{A P)~lA)*. Conversely, suppose m - 1 « G ((A - P)~lA)*, say (u/m) ■ (v/n) = (1/1) or 3/ G A -P. uul = mnl G A - P. Now u G P would imply uvl G P, a contradiction. Thus u € P, and ((.4 - .P)"l4)* = (A - P)~l{A - P). In

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other words, the non-invertible elements of the localization (A — PYXA form the ideal (A — P)~lP of M — PYXA. In general, a commutative, unital ring is said to be local if its non-invertible elements form an ideal. Since no multiple of a non-invertible element can be invertible, this is equivalent to saying that the non-invertible elements form a non-empty semigroup under addition. An alternative characterization is given by: Proposition 3.1.7. Let A be a commutative, unital ring. Then A is local iff iMax A\ = 1. Proof. Let x be an element of a proper ideal / of a local ring A. Then x is an element of the ideal A - A*, for otherwise 1 e xA QI and / = A. In particular, the unique maximal ideal of the local ring A is its ideal A - A* of non-invertible elements. Conversely, suppose Max A = {/}, and x,y &A - A*. Then xA ç / and yA ç /, since the proper principal ideals xA and yA are contained in some maximal ideal. Thus (x + y)A QxA +yA ç / c A, so that x + y e A - A*. Finally, note 0 c /
(3.1.6)

a +P*-> ( a / 1 ) +(A

- />)"'P);

-P)~lP

satisfies the universality property of Proposition 3.1.1 for the field of fractions of the integral domain A/P, so the residue field ((^4 — PYXA)/ l (iA - PY P) is this field of fractions. As a trivial example, note that each field is a local ring, and thus is its own residue field. EXERCISES 3.1A. Verify that the subtraction (3.1.2) is well-defined, and yields a /T-module structure M~XAK on M~XA so that (M~XAK, •, 1) is a .K-algebra. [Hint: For a non-computational verification of the associative aw for addition, use the injection exp : M X A -* A\; (m, a) <-* to give a monoid homomorphism (Af X A, +, 0) -» (A\, ■, 1).] 3.1B. Verify that / in (3.1.4) is a well-defined /C-algebra homomorphism with rj/ = / .

3.1C. Let Z be the set consisting of zero and the zero divisors of a commutative, unital ring A (cf. Exercise 1.4M). (a) Show that A - Z is a submonoid of A.

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167

(b) Show that the ring homomorphism TJ : A -* (A - Z)~lA of (3.1.3) injects. [The extension {A - ZYXA of A is called the total ring of fractions of the ring A.] (c) If A — Z is a proper submonoid of a submonoid M of A, show that the corresponding ring homomorphism 77 : A -* M~ lA of (3.1.3) does not inject. 3.1D. Let

-{[-: 3.1E. 3.1F.

3.1G. 3.1H. 3.11. 3.U.

3.1K.

3.1L.

n m,n 6 Z m

(a) Show that A is a subring of the ring Z 2 . (b) Show that A is an integral domain. (c) Identify the field of fractions of A. For a prime number p, show that the localization (Z - pZ)~lZ is the subring {m~1n\m Í pZ] of Q. Let AT be a real number, and let f(T)/g(T) be a rational function (quotient of polynomials) with g(x) # 0. Give a detailed proof that there is an open interval / containing x such that / -* R; t -> f(t)/g(t) is a continuous function. Let M be the submonoid {(1,1), (0,1)} of R2. Show that TJ :R 2 -> A/ -1 R 2 is the projection R2 -» R; (x, v) -* v. For / < ^ and (M, •, 1) <. (A, ■, 1), prove AT 1 / < A/ - U. Verify that (3.1.6) satisfies the universality property for the field of fractions of A/P. (a) For 17: A -* M~XA as in (3.1.3), consider the map TJ -1 : Spec (A/ - Û) -» Spec A of Exercise 1.4S. Show that TJ -1 is an injection, with image {Q G Spec A\Q n M = 0}. (b) For i* G Spec A, consider the map TJ,, : A -* (A - P)~lA of (3.1.3). Show that TJ^1 : Spec((^4 - P)~lA) -> Spec ^4 is an injection, with image {Q G Spec/i|ß ^ P}. Let jr be a real number. (a) Show that the ring Rx = {f(T)/g(T)\f(T\ g(T) G R [T], g(x) + 0} of rational functions defined at * is a local ring. (b) Identify the unique maximal ideal and the residue field of the ring Rx. Let A be the ring C(R) as in Example 3.3. Let x be a real number. Define a relation a on Sx = C(R) X [-<», x) X (x,*>] by (/, a, b)a(g, c, d) «* 3/, m e R U {-°o,00}. max{a, c) <. I < x < m <, min{¿>, d] and f\o,m) =g\a,m). (a) Show that a is an equivalence relation on Sx.

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(b) Defining R -* S?; r -+ (it -» r), -°o,«=)a, show that 5X° is an Ralgebra with (/, a, b)a + (g, c, d) a = ( / + g, max{a,c}, mm{b,d))a(c) Show that S" is a local ring, and identify its unique maximal ideal. (S" is known as the "algebra of germs of continuous real-valued functions at x") 3.2 Factors Let A be a commutative, unital ring. The group A* of invertible elements or units of A acts on A by (right) multiplication. Elements common to an y4*-orbit are described as associates. For example, Z* = {1, -1} s C2, and each integer n is associated with its negative. Non-negative integers are usually chosen as representatives for the Z*-orbits. The association relation in the ring A is a congruence on the monoid {A, •, 1), since A is commutative. Thus the set A/A* of orbits carries the quotient monoid structure, with A* as identity. Now suppose that A is an integral domain, so that A — {0} is a submonoid of (A, •, 1). Note that {0} is a singleton /l*-orbit. Then (A — {0})//4* is a submonoid of A/A*. A non-zero, non-unit element a of A is said to be irreducible if aA* = bA* ■ cA* => bA* = A* or cA* = A*. The set of irreducible elements of A is denoted as Irr A. In particular, if (A — {0})//4* is a free commutative monoid, then it is free on the set (Irr A)/A* of orbits of irreducible elements. [Interpreting {A — {0))/A* as a set of finite multisets, the elements of (Irr A)/A* correspond to the one-element multisets.] As motivation for the concepts of this section, consider the prototypical integral domain—the ring Z of integers. The Fundamental Theorem of Arithmetic says that (Z - {0})/Z* is the free commutative monoid on the set of orbits of prime numbers (cf. Exercise I 1.5F). The proof of the Fundamental Theorem of Arithmetic depends on the Division Algorithm. The Division Algorithm shows that, given a fixed non-zero integer d known as the divisor, the set Z¿ = {0,1 d - 1} of non-negative integers less than d is a loop transversal to the subgroup i/Z of (Z, +, 0). Thus a given integer n (known in this context as the dividend) has a decomposition n = n8d + ned as in (I 4.3.3), with the element n8d = dq e dZ and ned e Zd. The element ned of Z d is traditionally called the remainder and the integer q is called the quotient. Definition 3.2.1. Let A be an integral domain. (a) A is said to be a Euclidean domain, or ED, if there is a length function A -» N; a >-» \a\ such that: (i) \a\ = 0 *» a = 0; (ii) for each non-zero element d of A, there is a subset Ad of the set {a e^4||a| < |d|} of elements shorter than d such that Ad is a loop transversal in (A, +,0) to the principal ideal dA of A.

COMMUTATIVE ALGEBRA

169

(b) A is said to be a principal ideal domain, or PID, if each element of Id A is principal. (c) A is said to be a unique factorization domain, or UFD, if (A - {0})/A* is a free commutative monoid. D It is sometimes helpful to use the language of loop transversal codes in dealing with Euclidean domains. Consider a divisor, a non-zero element d of a Euclidean domain A. Imagine that A represents a noisy channel through which one wishes to transmit elements of the alphabet A. The elements are first encoded by the embedding L(d) : A -* A; a -* da, with image the code dA. Note that L(d) injects since A is an integral domain: dax = da2 =» d(ai - a2) = 0 =» ÛJ — a2 = 0. Now suppose that a dividend, an element n of A, is received as output from the channel. The errors to be corrected form the loop transversal Ad. The received word n is determined to have been subject to the error or remainder ned with n e dA + ned, so one has the error map ed : A -» Ad. Then n is decoded to n8d = n — ned. One obtains the decoding map Sd : A -» dA. Finally, the code element n8d = dq determines the quotient q = n8dL(d)~l as the original letter from A that had been encoded as dq. In any integral domain A, a non-zero element p is said to be prime iff pA is prime. This generalizes the familiar terminology from the ring Z, since a positive integer p is prime iff pZ G Spec Z. Note that, for non-zero elements a, d of an integral domain A, (3.2.1)

aA = dA «■ oA* = dA*.

Certainly aA* = dA* =* aA = aA*A = dA*A = dA. Conversely, suppose aA = dA, say a = du and a' = an. Then a = auw or a(l — vu) = 0. Since a is a non-zero and A is an integral domain, it follows that 1 — vu = 0, i.e. vu = 1. Thus « and u are invertible, so a G a'^4* and d G a/í*. Proposition 3.2.2. (a) In any integral domain A, prime elements are irreducible. (b) In a unique factorization domain A, irreducible elements are prime. Proof, (a) Suppose pA G Spec A - {{0}} and pA* = bA* -cA*. Certainly pA c bA and pA £ cA. Also be &. pA =* b & pA or c & pA, i.e. M £ p/1 or c4 ç pA. Thus M = p<4 or cA = /vl. If bA = /vl, then p/i* = bA* • ¿A* =» pA = bA ■ cA = pA • cA, so p = pea for some a in y4. Then p(l — ca) = 0 => ca = 1, so G4* = /4*. If C/4 = pA, then similarly M* = A*. (b) For /? e Irr A and ¿c G pA, say ¿>c = pa or bA* • o4* = pA* • aA*, one has bA* QpA* -A or c4* £p/l* -A, i.e. ¿ Gp/Í or c G pA D Let # [ r ] be a polynomial algebra. Define K[T] -> N; p ( D -»|/?(r)|=» if p ( D = 0 then 0 else 2deg /Kr) . The following result shows that, for a field K, the polynomial ring K[T] is a Euclidean domain with p(T) -» | p ( D | as

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length function. Recall that K[T] is already an integral domain if its coefficient ring K is. Proposition 3.2.3. Let K be an integral domain. Let d(T) in K[T] have invertible leading coefficient dm. Then V— (r(7)|degr(7) < m) is a loop transversal to d(T) • K[T]. Proof. By induction on the length |/(7)| of the dividend /(7), it will be shown that V contains a representative for the coset f(T) + d(T) • k[T]. If 1/(7)1 < 2 m , then / ( 7 ) = d(T) ■ 0 + / ( 7 ) G d(T)K[T] + V. Suppose V already represents cosets of polynomials of length less than the positive integer |/(7)| = 2" £ 2 m . Then f{T) - fnd¿ 7"- m d(7) G d(T)K[T] + V =» / ( 7 ) G d(T)K[T] + V. Finally, to show that V contains a unique representative for / ( 7 ) + d(T) • K[T], suppose / ( 7 ) = d{T)qx{T) + r,(7) = d(J)q2{T) + r 2 (7). Then m > deg[r1(7) - r 2 (7)] = deg{d(T)[q2(T) 9,(7)]} = deg d(T) + deg[ 92 (7) - 9 l (7)] = m + deg[ç 2 (7) - 9 l (7)] => deg[r1(7) - r 2 (7)] = deg[ 92 (7) - 9 l (7)] = - ~ =» ç,(7) = 9 z (7) and r 2 (7) = r,(7). D For an element k of an integral domain K, consider the monic polynomial (7 - k). Proposition 3.2.3 shows that K is a loop transversal to (7 - k)K[T\ Indeed, for any polynomial / ( 7 ) in K[T], one has / ( 7 ) = (7 - k)q(T) + f(k). Thus the principal ideal (7 - k)K[T] is the kernel of the ring homomorphism k[T] -» K; / ( 7 ) -*/(fe) onto the integral domain K. This shows that 7 — k is a prime element of K[T]. The relationship between the various domain properties of Definition 3.2.1 is given by the following: Theorem 3.2.4. (a) Every Euclidean domain is a principal ideal domain. (b) Every principal ideal domain is a unique factorization domain. Proof, (a) Let / be a non-zero ¡deal of a Euclidean domain A. The set |/ - {0}| of lengths of non-zero elements of /, as a subset of the positive integers, contains a least element \d\ for some fixed (but not necessarily unique) d in / - {0}. Certainly dA ç /. Conversely, for n in /, one has a quotient q e.A and a remainder r e.Ad such that n = dq + r. But r = n dq e /, so \r\ < \d\ implies r = 0. Thus n = dq G dA, and I Q dA. (b) Let y4 be a principal ideal domain. Each maximal ideal of A is of the form pA for a prime element p oi A. A mutually inverse pair fi,

(A-{0})/A*<=>(maxA)**

between (A - {0})/A* and the free commutative monoid (max A)*" of multisubsets of Max A will be exhibited:

COMMUTATIVE ALGEBRA

171

called multiplying out. For each one-element multisubset (pA) of Max A, define (pA)p = pA*. Note that /u, is well-defined by (3.2.1). Since (max A)*K is the free commutative monoid on these one-element multisubsets, (j. extends to a unique monoid homomorphism (also called fi) from (Max A)*K to the commutative monoid (A - {0})/A*. Injectivity of ¡J., namely the implication (pxA • p2A . . . pmA)*fi = (qxA • q2A . . . qnAYy¡. => (pxA • p2A ... pmAY = (qxA • q2A ... qnA)K, is proved by induction on m + n. The premise of the implication yields qxq2 ...qn + pxA = pxA in the field A/pxA, whence 31 <, i <, n. q¡A = pxA. Then (p2A . . . pmA)*fi = (... q¡_xA ■ qi+lA . . . » implies (p2A ... pmA)K = (... q¡_xA ■ qi+xA... Y by the induction hypothesis, whence (pxA • p2A . . . pmAY = {qxA ... q¡A ... qnAY, as required. Now consider a non-identity element aA* of (A - {0})/A*. Since the ideal aA is proper, it is contained in a maximal ideal pxA, say a = pxax. If axA is not proper, one has a, = p2a2 with p2A e Max A and aA c a,A c a2A. Continue thus. The process stops, since aA
* a,b e z\ of Z2 is a principal ideal

23

domain which is not Euclidean. [See W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN, Warsaw, 1974 (1st ed.), 1990 (2nd ed.), Chapter 3, Section 4.] D Example 3.2.6. If K is a field, then the polynomial ring K[T] is a Euclidean domain, by Proposition 3.2.3. Theorem 3.2.4(a) then shows that K[T] is a principal ideal domain. However, K[T, U] is not a principal ideal domain. Consider the ideal <7\ U)K[T,U\ of K[T, U], the ring kernel of the homomorphism K[T, U] ^ K; p(T, U) ~ p(0, 0) of (3.7). If
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Proposition 3.2.7 (Gauss'Lemma). The set X ofprimitive polynomials forms a submonoid of (K[T], •, 1). Proof. Suppose that the set of coefficients of the product of primitive polynomials p(T) and g(T) lies in a maximal ideal M of K. Consider the X-algebra (K/MXT']. By (3.4), there is a unique /C-algebra homomorphism ir : K[T] - (K/M\T\, T^T. Then (p(T)q(T)r = 0, but p(TY * 0 and q(TY * 0. This contradicts the fact that the polynomial ring (K/M\T] over the field K/M is an integral domain. D For the rest of this section, let K be a unique factorization domain, i.e. (K-{0))/K*= (IITK/K*)*K. Let X be the monoid of primitive polynomials of K[T], and let L = (K — {0))~1K be the field of fractions of K, The polynomial ring L[T] is a Euclidean domain, by Proposition 3.2.3, and thus a UFD, by Theorem 3.2.4. Recall (3.3): K[T]* = K* and L[T]* = L*. Theorem 3.2.8. The polynomial ring K[T] of a UFD K is itself a UFD. Moreover, (3.2.3)

Irr/i:[r] = I r r / ( : u ( Z n I r r L [ T ] )

for L = (K - {0])~1K and X the set of primitive polynomials ofK[T], Proof. Consider K c K[T] Q L[T]. The elements of Irr K have degree 0 in K[T]; since they have no non-trivial factorization in K, they can have none in K[T]. Suppose that, for an element f(T) of X n Irr L[T], one has f(T)K* =g(T)K* h(T)K* in K[T]. Then f(T)L* = g(T)L* ■ h(T)L* implies (without loss of generality) g(T)L* - L*, by the irreducibility of f(T) in L[T], so g(T) &K- {0}. Since f(T) is primitive, one then has g(T)K* = K*, as required. Thus lrr K V(X n Irr L[T]) ç KIT]. The set A" is a iC*-set (X,K*). Consider the construction of the free L*-set (X, K*)t LK\ = dX X L*)/K*, L*) over (X, K*\ as in Section I 2.3. Elements ( / ( D , k^^K* of (X X L*)/K* may be identified with elements (k¡/k2)f(T) of L[T], and each non-zero element of L[T] appears in this form. The bijection (X, K*)/K* -> ((X X L*)/K*)/L* = (L[T] - {0})/L* = (L[T] - [0})/(L[T])* of Exercise I 2.3F becomes a monoid isomorphism. Since L[T] is a UFD, it follows that X/K* is a free commutative monoid and (X - K*)/K* is a free commutative semigroup on the set (X n Irr L[T])/K*. Also (K - K*)/K* is a free commutative semigroup on the set (Irr K)/K*, since AT is a UFD. Thus [Irrtf U(.Yn In L[T])]/K* generates a free commutative subsemigroup of (K[T]/K*, • )—note that products of elements of Irr K all have degree 0, while products of elements of X n Irr L[T] all have bigger degree. Finally, the subsemigroup is in fact all of K[T] — K* [and in particular, equality holds in (3.2.3)]. Indeed, consider an element f{T) = *, • f\T) of K[T]. If the coefficients

COMMUTATIVE ALGEBRA

173

{/o. • • • »/deg/'} n a v e a common prime factor p2, set k2 = kxp2 and f(T) = k2f2(T). Continuing thus, one eventually obtains f(T) in the form f(T) = k,frO") with kr G K and / r ( D G X D Corollary 3.2.9. Fora UFD K, each multiuariate polynomial ring K[TV... ,Tn] is a UFD. Proof. By induction on n, using Theorem 3.2.8. D EXERCISES 3.2A. Prove that association is a congruence relation on the monoid {A, •, 1), for a commutative unital ring A. 3.2B. Determine the loop structure (I 4.3.4) given on the loop transversal ld to ¿Zin(Z, +,0). 3.2C. For an odd integer d, show that {r G Z||r| < \d/2\) is a loop transversal to a"l in (Z, +, 0). 3.2D. Determine the quotient and remainder when the dividend 2TS - 9T 3 + 97 - 2 is divided in Z[T] by the divisor T3 - IT + 1. 3.2E. Let K be an integral domain. (a) Prove, by induction on the degree of a non-zero polynomial f(T) in K[T], that f(T) has at most deg /(T) roots in K. (b) If K is infinite, prove that the ring homomorphisms (3.5) and (3.8) are injective. (c) Let K be a field. Can you prove the following: (i) K infinite «=» (3.5) injects; (ii) K finite «* (3.5) surjects? 3.2F. Show that the ring A of Exercise 3. ID is a Euclidean domain, with the determinant as length function. 3.2G. Let a be a non-zero element of a principal ideal domain A. For each maximal ideal P of A, let eP be the multiplicity of P in the factorization aA*

[^-{[Î

~1]|»'H

174

CHAPTER II

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(a) Show that Z[j(b) Show that

m\ is a subring of Z|.

>-»

mib a [ib a

is a unital ring homomorphism. (c) Show that N : (Z[v/-m], • , / ) - > (N, -, 1); / -* det fJ is a monoid homomorphism. (d) Show that ZU - m ] is an integral domain. (e) For m = 2, show that Z\i—m is a Euclidean domain. (f) For m = 5, show that Z [ / - m ] is not a unique factorization domain. 3.2L. Show that the polynomial ring Z[T] is a UFD, but not a PID. 3.2M. Express the polynomial 12T2 + 6T{T2 - 36T22 as a product of primes

inZ[T„r 2 ]. 3.3. Modules over Principal Ideal Domains A module V over a ring S is said to be finitely generated if it is spanned by a finite multisubset. The main result of this section, Theorem 3.3.4, identifies the isomorphism classes of finitely generated modules over a PID A. Specifically, for a positive integer m, an {m - D-simplex in a poset {X, <.) is an m-element multisubset (xx xm) of X with xx ^ x 2 ^ ... £xm. The set of (m - l)-simplices in (X, ^ ) is denoted by (X, ^ )„,_,. Then the isomorphism classes of A -modules spanned by m-element multisubsets correspond to (m - l)-simplices in (Id A, 2). For Euclidean domains A, an algorithm for determining the simplex of a module is given: The algorithm essentially consists of Gaussian elimination. Taking A as the ring Z of integers, the algorithm classifies finitely generated abelian groups. In the following section, the algorithm classifies finite-dimensional linear dynamical systems by taking A as the polynomial ring K[T] over a field K. The special property of principal ideal domains that facilitates the classification of finitely generated modules is given by the following characterization. Theorem 3.3.1. A commutative, unital ring S is a principal ideal domain iff each submodule of a free S-module of finite rank is a free module of equal or lesser rank. Proof. First assume that submodules of finitely generated free modules are free modules of equal or lesser rank. Since {0} = {0}274 is a free {0}submodule of the free {0}-module {0}97, the ring 5 is not the zero ring {0}. Moreover, a zero divisor s of S would yield an 5-submodule sS of Ss that

COMMUTATIVE ALGEBRA

175

was not free [cf. Example 2.3.3 (a)], so S is an integral domain. Finally, a non-zero ideal / of S, as a submodule Is of the free S-module Ss of rank 1, is a free module of rank 1, and thus a principal ideal. Conversely, suppose that 5 is a principal ideal domain. Then submodules of free modules of rank 1 are free modules of rank 1 or 0. Fix a positive integer n. Suppose, by induction, that submodules of free modules of rank r < n are free of rank at most r. Let U be a submodule of S". Consider the projection p: S" -» S; (slt..., s„) -► sn, with kernel Kerp = S"~l. If U <. Ksip, then U is free of rank at most n — 1, by the induction hypothesis. Otherwise, Up is a rank 1 free submodule of S, again by the induction hypothesis. Also KerCply), as a submodule of the free rank (n - 1) module Kerp, is free of rank at most n - 1. Since Up is free, the exact sequence 0 -» KeKply) —» U —» Up —► 0 splits, and then U, being isomorphic to the direct sum Keripli/) © Up, is free of rank at most (n - 1) + 1 = n. D Let Us and Vs be right modules over a ring S, with corresponding groups Aut Us and Aut Vs of automorphisms (invertible module endomorphisms) under composition. Thus U is a right (Aut i/)-set and F is a right (Aut K)-set. Now consider the set S(U, V) of S-homomorphisms from U to V. This set is a right (AutC/)-set under the action Ö:/-» Ö"1/ and a right (AutF)-set under the action . Thus it is a right (Aut U) X (Aut F)-set under the action (0,- The (Auti/) X (Aut F)-orbits on SW,V) are known as equivalence classes: Two S-homomorphisms ft,f2 are equivalent if 36 e Aut U. 3

)„_!, for n = min{/, m), classifies the equivalence classes of .¿-homomorphisms from a free Amodule U of rank / to a free A -module V of rank m, for positive integers / and m. This will be done by induction on / and m. The base case 1 = m - \ is immediate from (3.2.1) and the observation Aut AA = {A -» A; 1 -* u\u G A*): The equivalence class of the typical /4-homomorphism L(d): A -* A; 1 -» d then corresponds to the 0-simplex <<¿4> in (Id A, 2). Note that an (n - l)-simplex ¿¡A od2A a ••• ~2dnA determines the (well-defined) equivalence class of the ^4-homomorphism

¿¿K) ©0:^'- Í (BA\ © j ®A\ -+1 (BA\ = Í ¡©^JffiOisí®A\ © Í " £ M ) =Am. Proposition 33.2. Let U be a module and let V be a free module over the principal ideal domain A. Then: V0*f(=A(U,V),3dl f = L(d1)f1 and

e ^ . a / j eA(U,V).3d1 ImiMJ-A.

e V.

176

CHAPTER II LINEAR ALEGEBRA

Proof. Recall the factorization

V02 e K\ u,/0 2 /i <; d,^.

Indeed, suppose cL4 = uxfOxA + uxfd2A, say d = uxf8xax + uxf62a2 = i/(0i a i + 0 2 a 2 ) e U ( € r Im(/0). Then d, e dA implies that dA*

= dxA*

M

(3.3.2)

Vu2 e [/, u 2 / M <; d ^ .

Indeed, suppose cA = uxfdxA + u2fOxA, say c = uxfOxax + u2fOxa2 = (M,a, + u2a2)f6x e Ufl6>» Im(/0). Then cA = d,/4 by the argument yielding (3.3.1). Finally, Vw2 e t/, V02 e F', u 2 /0 2 .4 <; dxA.

(3.3.3)

Certainly 36 e A «j/flj = Uj/0,6, by (3.3.1). Consider 0J = 02 - 0,è + dx. Then ^ = uxf6x = u,/0;, and K 2 / 0 2 = u2(b - l)f0x + u2f0'x. By (3.3.2), one has u2(b - Df0xA ^ dxA. By (3.3.2) with 0J replacing 0,, one has u2fd'xA ^ dxA. The containment (3.3.3) follows. Let ß : I -» F be a basis for K For each element i of /, the isomorphism (2.3.7) yields a functional

h:V-+A;v~i{vR^yx) selecting the i-th coefficient of a module element v with respect to the basis ß. By (3.3.3), one may then define /, e A(U, V) such that, for u e U, one has ufßi = dx(ufxßi) for i e / . Thus f=L(dx)fx. Also dx = uxfex = uxdxfxdx = dx(uxfx9x) implies M , / ^ , Sy4*, so lm(fx6x) = A. D Proposition 3.3.2 is used to complete the inductive proof by showing that each ^4-homomorphism / : U -» V from the free module U of rank / to the free module V of rank m is equivalent to Y,"_xL(d¡) for some (« - l)-simplex dxA?d2AD ■■■ 2dnA. If / = 0 , then dxA = ••• = dnA = {0}. Otherwise, Proposition 3.3.2 yields uxeU,fx e.A(U,V) and 0j e V such that ¿(«i)/i*i =

(3-3-4)

^

L(Ul)

COMMUTATIVE ALGEBRA

177

/I'M

Note that the exact sequences 0 -» Ker/jfl, —► U —* A —► 0 and 0 —► K e r f l j — * V - ^ A — » 0 split, namely U = ALiux) © Ker/,0, and V = AL(ul)f1 © Ker 6X. Moreover, fx restricts to an /1-homomorphism f[ : Ker/,0! -» Ker0,. By induction, there is an in - 2)-simplex e2A 2 e3A 2 •• 2enA such that f[ is equivalent to T,"_2Lie¡). Also, fx restricts to an isomorphism /," : ALiux) -* ALiux)fx. Thus / , is equivalent to L(l) © E?_ 2 ¿ (e ( ):í/-> V, whence f = Lidx)fx is equivalent to Z%xLid¡) with d¡ = dxe¡ for 2 <. i <, n and with dxA 2 d2A 2 ■•• 2 dnA. Summarizing: Theorem 33.3. Let A be a principal ideal domain. Let U and V be finitely generated non-zerofreemodules over A, of respective ranks I and m. Then for n = min{/, m], the map (Id A, 2)„_i ^A(U,V)/(AutUX

(dxA 2 ••• 2d„/4)-»

AutF);

¿ L K - ) (Auti/X AutF)

bijects. D Now let W be a non-zero finitely generated module over the principal ideal domain A. Suppose that W is spanned by ß : Z m -» W, so the y4-homomorphism Ri ß ) : Am -» W [an instance of (2.3.2)] surjects. By Theorem 3.3.1, Ker R(ß) is a free /I-module of rank /, say, with 0 :£ / ^ m. If / = 0, then ß is a basis for W, which becomes free of rank m. In this case W = Am s sfLjU/d,/!) with dj/i = d 2 ^ = ••• = dmA = {0}. Otherwise, Ker/?(/3) is a non-zero free submodule U of the free module V = Am, and (2.3.2) yields the exact sequence (3.3.5)

0 — U-Uv^^W-^ 0.

In general, a /ree resolution of a module if is an exact sequence ••• -* An -* •■■ -* A2 -* Ax ^> W -* 0 oi modules with Ax, A2,..., An, •■• all free. By the First Isomorphism Theorem applied to Riß) in (3.3.5), one has W s V/Ker Ri ß) s V/Uf, so the isomorphism class of W only depends on the equivalence class of / in (3.3.5). By Theorem 3.3.3, / is equivalent to E(.jL(d ( ) for an (/ - l)-simplex dxA 2 — 2 d,A in (Id A, 2). Thus W = Am/idxA © ••• © d,A) = (®¡_x(A/d¡A)) © Am-'. Suppose dk+xA = ^k+2^ = '" = d¡A = {0} for some k with 1 <, k < I (by convention k = I means d,A D {0}). Then W= ®f_xiA/d¡A) ®Am~k. If dxA = A, then ( C s ®Í_2iA/d¡A) ®Am~k could have been spanned by some ß' = Im_x -» W. One thus obtains the following classification of isomorphism classes of finitely generated .¿-modules. Theorem 3.3.4. Let A be a principal ideal domain. Consider the poset (Propel, 2 ) of proper ideals of A. Thenfinitelygenerated non-zero A-modules

178

CHAPTER II LINEAR ALEGEBRA

W are classified up to isomorphism by the set U n 6 N (Prop A, 2)„ of simplices in (Prop A, 2), namely Ws

(3.3.6)

/or.,4 D d , ^ 2d 2 -^ 2 "' 2 d m ^ -

© {A/diA)

i-l

D

In the context of (3.3.6), suppose dkA 3 dk+1A = 0 for some A: e {0,1,..., m). Then the natural number m - k is called the ranfc of W. The ideals dxA,...,dmA are called the invariant factors of W. A module of rank 0, and the zero module itself, are called torsion modules. Suppose that d is a non-zero, non-unit element of A, such that dA* has the factorization dA*

-* (a + p\lA,...,a+ p'rA) bijects. For a non-zero torsion ^-module W with (m - l)-simplex dxA "2.d2A 2 ••• 7>dmA, suppose dmA*(p — ((p,i4)««-...(prA)e"")K with \{piA,...,prA\| = r. Then for 1 <.j'< ¿ m, one has djA*

(3.3.7)

W=é

i-l

©(¿/pfM).

y-i

The decomposition (3.3.7) is called the primary decomposition of the torsion module W. The proper ideals p'n A in (3.3.7), i.e. with c i; > 0, are called the elementary divisors of W. If r = 1, then W is said to be a primary module, or more specifically a p^-primary module. If A = Z, so that ^ is a finite abelian group, then the p^-primary modules ®JLx(A/p''iA) in (3.3.7) are the Sylov subgroups of W. More generally, abelian groups of prime-power order are primary Z-modules. For the rest of the section, assume that A is a Euclidean domain. Given an element [a b] of A X (A — {0}), a generator d of the principal ideal dA = aA + bA may be found by the Euclidean Algorithm begin [n0d0] ■■= [ab], i » 0 while «¡Cj * 0 do \.ni+ldi+l]-= [n¡d¡] i -=i+l end while output d •■- d¡

-n,SélL(d,yl

-i

l

[¡i]

COMMUTATIVE ALGEBRA

179

described using the loop transversal code terminology introduced after Definition 3.2.1. Note that n (+1 = d¡ and di+l = n¡ - d¡(n¡8d L(d,) - 1 ), so ni+lA + di+1A <, n¡A + d¡A. Conversely, n¡ = ni+l(n(8diL(di)~l) + di+l and d¡ = n¡+1, so n¡A + d¡A :S ni+1A + di+1A. If n¡ed¡ = 0, then n¡A + d¡A d¡A. Also aA + bA = n0A + dQA. Thus aA + bA = dA. The algorithm terminates in at most \b\ steps, since di+l &Ad implies 0 < \d\ < ••• < |d i+1 | < \dt\ < < |d0l = U Given an element / of A" or A(A', Am) with n = min{/, m), the Euclidean Algorithm combines with Gaussian elimination to determine the (n - l)-simplex dxA 2 ••• 2d„A such that / is equivalent to E^ZXi/,). Note that column operations are part of the action of Aut Am on A(Al, Am). Similarly, one obtains row operations as part of the action of Aut A' on A(Al,Am) by premultiplying by permutation, reseating, and shear matrices. Now the 2 x 2 matrices in the while loop of the Euclidean Algorithm are a shear matrix and a permutation matrix respectively. Using the Euclidean Algorithm, one may apply column operations to reduce / to an equivalent matrix / ' with zero off-diagonal elements in the first row, such that f'uA = fnA + •■• +flmA. Similarly, one may then apply row operations to reduce / ' to an equivalent matrix / " with zero off-diagonal elements in the first row and first column, such that f'[xA = fnA + ••• +fimA + f2iA + ■■• +fnA. If VI <, i <, I, VI f£ ; £ m, f"j G fûA, one may then take dx = f"n. Otherwise, if fab ^ f\\A, one may use a shear to add the 6-th row of / " to the first row, and then reiterate the use of the Euclidean Algorithm. Eventually / is reduced to an equivalent matrix R(dx) + g with g in A(Al~l,Am~*\ and such that VI ^ i' ^ I - 1, VI £j ¿ m - 1, g(; G dxA. By induction, g is equivalent to E ^ L i e , ) for an (n - l)-simplex e2A D ••• oenA. Setting d, - dxe¡ for 2 £ i £ n, one thus obtains / as equivalent to L"mlL(di) for the (n - l)-simplex dxA~2. d2A 2 "• ~2.dnA. This procedure determines the simplex of a non-zero, finitely generated module W by operating on the homomorphism / in a corresponding free resolution (3.3.5). Example 3.3.5. Consider 2 -2 6

8 -2 24

-4 4 -12

4 -4 e 12.

n-

nt to 2 -2 6 =

8 -2 24

-4 4 -12

2 -2 6

0 6 0

"l 4' -4 0 12. 0 0 0 0' 0 0 0 0

4 1 0 0 i

2 0 1 0

-2 0 0 1

180

CHAPTER II

LINEAR ALEGEBRA

ind then to 1 1 -3

0 1 0

2 0' 0 -2 1 6

0 6 0

0 0 0

0" '2 0 = 0 0 0

0 6 0

0 0 0

0 0 0

EXERCISES 3.3A. Show that the submodule 21[T] + TZ[T] of the free Z|T]-module Z[T]Z[T] is not free. 3.3B. In Proposition 3.3.2, suppose that ß : Z, -» U and y : Z m -* V are bases, so that R(ß)fR(yYl eA(A', Am)=A'¡'. Show that each en1 try of the matrix R( ß)fR(y)~ lies in the ideal dxA of A. 3.3C. Let U be a submodule of a (not necessarily finitely generated) free module V over the principal ideal domain A. Let ß : / -> V be a basis for V. Let X = {(J,S)\J 2 1 , S ç U, S a basis for U D Im R(ß\j)}. Define a partial order ^ on X by ( / j . S , ) ^ (J2, S2) <=> /j 2 J2 and 5, ç 5 2 . (a) Show that (X, ^ ) is an inductive poset. (b) Use Zorn's Lemma to obtain a maximal element (/, S) of (X, <). If 3 i e / - /, use the embedding (U n Im R(ß\ 7 U ( 1 ) ))/ ([/ n Im Ä(/3|7)) -♦ Im Riß \, u W )/Im Ä(/31 y ) s A to obtain a contradiction to the maximality of (/, 5). (c) Deduce that every submodule of a free y4-module is free. 3JD. (a) Let W be a module over the principal ideal domain A. An element x of W is a torsion element if the annihilator An ^.£4 is non-zero. Prove that the set of torsion elements of W is a submodule Tor W of W. (b) Let If be a non-zero, non-free, finitely generated y4-module, with simplex dxA 2 ••• 2d m ^4, where dkA ^> dk+xA = 0 for some fc e {1 m}. Prove that Tor If is a non-zero, non-free, finitely generated A -module, with simplex dxA 2 ••• ~2dkA. 3.3E. Determine the invariant factors of the finite abelian group with multiset <4Z, 4Z, 25Z, 27Z, 27Z, 29Z> of elementary divisors. 3.3F. (a) Show that each finite abelian group is the direct product of its Sylov subgroups, (b) Show that H=

1 a 0 1 0 0

¿>]

c a,b,c

lj

e

COMMUTATIVE ALGEBRA

181

is a non-abelian subgroup of SL3(Z6) (cf. Exercise 2.4L). Show that H is the direct product of its Sylov subgroups. (c) Exhibit a finite group that is not the direct product of its Sylov subgroups. 3.3G. Prove the Chinese Remainder Theorem: Let A be a principal ideal domain. Let px,...,pr be distinct prime elements of A. Then for natural numbers ex,...,er and for d = p'xl...p'r, there is an y4-algebra isomorphism A/dA

3.3H. 3.31. 3.3J. 3.3K. 3.3L. 3.3M.

s

X\A/p\>A

i-i

(cf. Exercise O 4.4J). Determine the primary decompositions of the groups of units of the rings Z21 and Zl0S. For the abelian group Z 4 /([l - 1 1 -1]Z + [002 -2]Z), determine the invariant factors. Show that there are 7 isomorphism classes of abelian groups of order 32. (Hint: Cf. Exercise I 1.5G.) Exhibit a free subgroup of n „ e N(Z/2nZ). Are the groups Z50 © Zg0 and Z40 © Z100 isomorphic? (a) Set B0 = n„ e N (Z/4Z), Bx = n, i e N ((Z/4Z) © (Z/2Z)), and B2 = n„ 6 N (Z/2Z). Show that B0 X Bx s B0 X B2, but Bx SÉ B2. (b) If BÜ,BX, B2 are finitely generated abelian groups, prove BQ x Bx = B0 X B2 =* Bx = B2.

3.4. Linear Dynamical Systems According to Section O 4.2, a dynamical system is a set X with a function T : X -* X, the evolution operator. In other words, A!" is a -set for the free monoid (T) on the single generator T. Analogously, a linear dynamical system (U, t) is an 5-module U together with an S-homomorphism / : Us -» Us, for a ring 5. The evolution operator t is an element of the ring End s U of 5-endomorphisms of the 5-module U. Two such endomorphism tx,t2 are said to be similar if there is an action morphism (0, tx <-* t2) from the -set ([/, <*!» to the
182

CHAPTER II LINEAR ALEGEBRA

dt2 = ijö in End s U. Alternately, one may consider the action 6 : t <-* 8~1t6 of Aut Us on End Us. Then tx and t2 are similar iff they lie in the same (Aut i/)-orbit. The fundamental problem in the study of linear dynamical systems is to classify their similarity classes. The ideal is to have an algorithm deciding whether or not two given linear dynamical systems are similar. For a commutative unital ring K, consider a linear dynamical system {V, t) in which F is a unital ^-module via the representation K -* End Vz. This representation makes End VK a K-algebra. The universality property (3.4) for the polynomial algebra K[T] then yields a unique AT-algebra homomorphism K[T] -» End Vz; T -* t representing V as a £[r]-module. Note that two tf-endomorphisms tut2 of V are similar iff (K, /j) and (V, t2) yield isomorphic X[T]-modules. If F is a finitely generated K-module, then it may be expressed as a cokernel of K[r]-modules. Theorem 3.4.1. Let Vbe afinitelygeneratedfreemodule over the commutative ring K, with basis ß:Z„-* V. Let V be a K[T\-module via K[T] -» End VK\ T <-* t. Then there is an exact sequence (3.4.1)

K[T]"

r

-*(W)"'.

^

r

^

^

O

.

Proof. As a free ^-module with basis ß : 1n -» V, the module V is isomorphic with K" via R( ß ) : K" -» V. Thus ( 6 End ^ yields R(ß)tR(ß)-1 e End K" = K(K", Kn) z K[TlK[T]n, K[T)n). Moreover, the iC-isomorphism Riß): K" -» V yields a surjective jRT[r]-homomorphism Riß) : K[T]n -> V. The elements T and R(ß)tR(ß)-* of K[TlK[T]n, K[T]") yield the same maps V -» V, so Im(r R{ß)tR{ß)~l) <> Ker(R(ß):K[T]n - V). Conversely, let X = Ker(/?(0): K[T]" - V)/ lm(T - Riß^Riß)-1). Suppose X is non-zero. Let (p¿T) P„(T)) + Im(r - i?(/3)i/?(j3)-1) be a non-zero element of X, represented by an element (p^T),...,pn(T)) such that max{degp^T),...,degp n (T)) = m is minimal amongst all such representatives. Note that m is positive, since KeK J?( ß) : Kn -» V) = {0}. Set p ( (D = E,% p y r>. Then (p,(T),.... ptt(T)) + I m ( r - R(ß)tRiß)-1) = (EJLVPI/T' EiVoV-yrO + (/»im. • • •. Pnm)Riß)tR{ßr'Tm~l + lm(T - R(ß)tR(ß)-1), contradicting the minimality of m. Thus (3.4.1) is exact. D Corollary 3.4.2 (The Cayley-Hamilton Theorem). The determinant det(r - R(ß)tR(ß)~l) lies in the annihilator AnK[T]V. Proof. Set f=T-R( ß)tR(ß)-1 e K[TlK[T]n, K[T]n) = K[T]nn, so that F = CokeK/: K[Tf - tf[r]")Jby Theorem 3.4.1. Now by (2.4.7) and Theorem 2.4.3(b), one has / / = (fTfT)T = \fT\ = |/|. Then for p e tf[r]", one has (p + I m / ) | / | = p\f\ + I m / = / > / / + I m / = I m / . D

COMMUTATIVE ALGEBRA

183

Note that dct(T - RißMiß)'1) = det[R(ßXT- t)R(ß)~l] = det(T - t) in Corollary 3.4.2. For a square matrix t or endomorphism í of a finitely generated free module, the monic polynomial det(T - i) is called the characteristic polynomial of t. Example 3.4.3. Consider the complex number A:

t=

-y

y X

II

as in Exercise 2.4Q. Then ÁT~x ~y = T2-2xT+(x2+y2). det „ y T—x Thus t is a root of the real polynomial T2 - (f + t)T + \t\2 e U[T]. D For the remainder of this section, let K be a field, so that K[T] is a Euclidean domain. Let (V, t) be a linear dynamical system of finite dimension n. Note that the exact sequence (3.4.1) becomes a free resolution 0 —» K[T]n -£♦ K[T]n - ^ - » V —► 0 of the K[T]-module V, an instance of (3.3.5). By Theorem 3.3.4, there is an (m - l)-simplex dxK[T] 2 d2K[T] 2 •• 2 dmK[T] in (Prop K[T], 2 ) yielding an isomorphism A

p : £ K[T]/d,K[T]

(3.4.2)

— V

«-i

of /C[r]-modules. No d¡ can be zero, since K[T] is infinite-dimensional as a K-module. By (3.3) and (3.2.1), the polynomials dx dm may be chosen uniquely as monic polynomials. These polynomials are then called the invariant factors of t, and the isomorphism p of (3.4.2) is called the rational canonical form of t. Note that the algorithm of Section 3.3 may be used to determine the monic polynomials dv...,d„. Now d^lT] = KeKK[T] -» End Vz; T -* t), the annihilator of the K[T]-modale V. The monic polynomial dx is called the minimal polynomial of the endomorphism t. For a monic polynomial p(T) in K[T], the quotient K[T]/p(T)K[T] is a vector space over K with basis {T'lO <, i < deg p(T)}. The .KTrl-homomorphism L(T) : K[T]-* K[T]; q(T) *-> Tq(T) induces a ÍL-endomorphism UT)p of K[T]/p(T)K[T]. Note that the annihilator AnK[T]K[T]/p(T)K[T] is p i D t f t r ] . Now the degree of the monic characteristic polynomial det(r - L(T)p) of UT)p is deg p(J), so the Cayley-Hamilton Theorem shows that this polynomial is just p(T). Returning to the rational canonical form (3.4.2), the endomorphism E™_ ] L(T)d of the domain may be expressed as 1 1 ( 1 2 * ^K[T]/d)K[T]\ ® L{T)cli v © I ¿, ^KlT]/d,K[T]\

' - 1 \v;'
'

;>i

'

184

CHAPTER II

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whose characteristic polynomial is nT~id¡. Since VK has basis m

(3.4.3)

p : U {P + d,tf[r]|0 £ ; < deg d,} -» F, i-i

the characteristic polynomial of f is d e t ( r - R(p)tR(p)~l) E J L i ^ ' V ) = n?-id>- Summarizing:

= det(r -

Theorem 3.4.4. Let Vbe a vector space of finite dimension n over the field K. Then endomorphisms of Vare classified up to similarity by their invariant factors dvd1,...,dm, with each factor a multiple of its predecessor. The minimal polynomial of an endomorphism is itsfirstinvariant factor, while its characteristic polynomial is the product of its invariant factors. D Example 3.4.5. For a positive integer n and field K, elements of the group GLn(K) are conjugate iff they are similar. Thus conjugacy classes in GL„(/0 are classified by the invariant factors of their members. Recall (cf. Exercise 2.4M) the isomorphism of GL2(Z2) with the symmetric group S3 given by restricting the action (Z2)J x (Z 2 )| -» (Z2)2 of (1.3.4) to ((Z2)f - {[0,0]}) x GL2(Z2) -» ((.I2)2 - {[0,0]}). Recall Proposition I 3.2.2 classifying conjugacy classes in ((Z2)j - {[0,0]})! by partitions of 3. Then the S3-class l 3 corresponds to invariant factors (T + 1),(7 + 1). The 53-class ü'21 corresponds to the single invariant factor T2 + 1. The S3-class 31 corresponds to the single invariant factor T2 + T + 1. D As a K[T]-modale, the linear dynamical system (V, t) is a torsion module, and thus is the codomain of a ÍC[r]-isomorphism r

(3.4.4)

(: ©

¿-i

m

© K[T]/pf»K[T]

j - \

- V

yielding a primary decomposition, as in (3.3.7). Choosing the prime elements p, to be monic, the isomorphism Ç is called the Jordan canonical form of the ÜT-endomorphism t. (Classically, this nomenclature has been reserved for the special case where all the p¡ are linear.) The monic polynomials pf'i in the Jordan canonical form are called the elementary divisors of t, and the characteristic polynomial of t is the product rij«. 1 nyl 1 p'" of its elementary divisors. Endomorphisms of V are classified up to similarity by their Jordan canonical forms, just as they are classified up to similarity by their rational canonical forms according to Theorem 3.4.4. Suppose that p¡ is a linear polynomial T - A,-. Then Ker(i - A,) is called the eigenspace of t corresponding to the eigenvalue \¡, and non-zero elements of Ker(f - A,) are called eigenvectors of t belonging to A,. (The synonymous adjectives "proper" or "characteristic" were preferred to "eigen-" in English-speaking countries

COMMUTATIVE ALGEBRA

185

during or immediately after times of war with German-speaking countries. However, these synonyms are so overworked—"proper subspace", "characteristic polynomial", etc.—that their use is best avoided.) Note that the multiset (e¡j\l £j 0) yields a partition of EyL^-. Its length dim Ker(f - A,) is called the geometric multiplicity of the eigenvalue A/( while its sum E™_i«y is called the algebraic multiplicity of A,. Note that the eigenvalues of t are just the roots of its characteristic polynomial. Example 3.4.6. Pick a scalar A e i , and consider the polynomial p(T) = ( r - A)e for e > 0. The endomorphism L(T)p of K[T]/p(T)K[T] has a 1-dimensional eigenspace spanned by (T - A)' -1 + p(T)K[T]. Thus L(T)p has eigenvalue A with geometric multiplicity 1 and algebraic multiplicity e.

□

Example 3.4.7. Consider the vehicle suspension system of Example 2.2.2, described by the differential equation (2.2.4) with corresponding homogeneous equation ks + Is' + ms" = 0. This equation may be rewritten in the form [s's"] = [ss']

Consider the R-endomorphism

0 1

0

1

-k/m -l/m

-k/m

-l/m j of R2. The qualitative behavior of

the suspension system is described by the number of eigenvalues of this endomorphism. If there are no eigenvalues, then the differential equation has oscillatory solutions, suggesting that the vehicle body will bounce up and down relative to the axle. Subjectively, the ride is "soft" or "wallowing." If there are two eigenvalues, then the suspension system is slow to respond to displacements of the axle. Subjectively, the ride is "hard" or "rough." The ideal behavior occurs when there is a single eigenvalue (with algebraic multiplicity 2). The suspension absorbs irregularities of the surface over which the vehicle is traveling, but without setting up oscillations. D EXERCISES 3.4A. Let A be a 2 X 2 real matrix. Show that A2 is a real linear combination of A and / 2 . 3.4B. Show that each quaternion (cf. Exercise 2.4S) is the root of a real quartic polynomial. Factorize the quartic. 3.4C. Let K be a commutative, unital ring, and let / be an n X /i-matrix over K. Show that the characteristic polynomial of / is of the form j " _ 7 « - i t r f+ ... + ( _ 1 ) n d e t ^

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3.4D. Give a direct proof that the characteristic polynomial of L(T) e End(K[T]/p(T)K(T))K is p(T\ for the monic polynomial p(T) over a field K, without using the Cayley-Hamilton Theorem. 3.4E. For a field K, show that the invariant factors of elements of GL„(K) all have non-zero constant term. 3.4F. Classify the conjugacy classes in the groups GL3(Z2) and GL2(Z3). 3.4G. Show that elements of SLn(K) are conjugate in SLn(K) iff they are conjugate in GLn(K). 3.4H. Classify the conjugacy classes in SL2(Z3). 3.41. Fix a positive integer p. Let wp denote rotation of the Cartesian plane R2 counterclockwise by an angle 2ir/p. (a) Show that ta is an element of End R R 2 = R\. (b) Show that tap is a complex number (in the sense of Exercise 2.4Q). (c) Show that tap is a root of Tp - 1 in C[T]. (d) Show that Tp - 1 = YlfjJiT - ta'p) in C[r]. (e) Show that the subgroup of GL2(IR) generated by Ç~x as an element of Dom Ç. (Hint: Use Exercise 3.41.) 3.4K. Solve the homogeneous differential equation ks + Is' + ms" = 0 of Example 3.4.7, taking the constants k, l, m in intervals representing the suspension system of Example 2.2.2. In particular, identify the three different cases corresponding to the three possible cardinalities of the set of eigenvalues of

_¡m

e EndflR2)R.

3.4L. Let C(R, C) denote the complex vector space of continuous functions M: U -* C (with componentwise operations). Let D(R, C) denote the subspace of C(U, C) consisting of continuously differentiable functions. Define A : D(U, C) -> C(R, C); u -» u'. For a finite subset A of C, let E\ denote the subspace of D(R, C) spanned by {e^U e A}. Determine the rational canonical form and the Jordan canonical form of (E\, A). [Note: With suitable time units, light whose spectrum comprises the frequencies o>,,..., o>n is represented by elements of EA for A = {i&>],...,i&>„}. The elementary divisors of the C[r]-module £A

COMMUTATIVE ALGEBRA

187

are the elements (T - iw^lT], ...,(T- i w„)C[T] of Spec C[T]. This may be the origin of the term "spectrum" for the set of prime ideals of a commutative, unital ring.]

3.5. Elementary Field Theory Throughout this section, let K be a field. If A is a ii-algebra, then one may use the injection K -* A; k >-* Ik (cf. Exercise 3A) to identify K as a subalgebra of A. Given a non-zero element t of A, consider the /C-algebra homomorphism K[T] -» A; p(T) -* p(t) of (3.4), with image K[t] — {p(t)\p(T) G K[T]). If this homomorphism injects, then / is said to be transcendental over K, and the homomorphism yields a ^-algebra isomorphism of K[T] with K[t]. Otherwise, the homomorphism has a non-trivial kernel of the form p(T)K[T] for a uniquely specified monic polynomial p(T). In this case t is said to be algebraic of degree deg p(T) over K, and p(T) is called the minimal polynomial of the algebraic element t. [If p(T) is quadratic, cubic, etc., then t may be described respectively as quadratic, cubic, etc.] Complex numbers that are algebraic over the rationals are known as algebraic numbers. Thus 3^2 and i are algebraic numbers, while e and •flare transcendental. The quaternions i,j, and k are quadratic over R. In general, the K-algebra A is said to be algebraic over K if each non-zero element of A is algebraic over K. Let / be an algebraic element of A over K, with minimal polynomial p(T). One may identify t with the endomorphism RK(t) :a >-* at of the X-vector space K[t] of dimension deg p(T). The First Isomorphism Theorem applied to (3.4)yields the rational canonical form p : K[T]/p(T)K[T] -» K[t] of t. Thus p(T) is both the minimal and the characteristic polynomial of the /¡¡T-endomorphism t. Motivated by Exercises 2.4Q and 3.4C, one defines (_ l)deg Pp(ff) t o b e t h e norm d e t R^t) Qf t over K a n d t h e negative 0 f t n e coefficient of T(deg P ) _ I in p(T) to be the trace tr RK(t) oft overK. Thus the norms of the complex number or quaternion i over R or Q are all 1, while their traces are all 0. Proposition 3.5.1. Suppose that the K-algebra A has finite dimension as a K-vector space. (a) Each element t of A is algebraic overK, of degree not exceeding dim AK. (b) If A has no zero divisors, then it is a skewfield. Proof, (a) The N-tuple N -* A; i -» f is dependent, so K[T] -> A; q(T) -* q(t) has a non-trivial kernel p(T)K[T], and t is algebraic. Also deg p(T) = dimOC[T]/p(T)K[T])K ^ dim AK. (b) Let t be a non-zero element of A. By Proposition 2.4.8 or Theorem 2.4.9(c), the injective .K-homomorphisms R(t) : A -» A; a -» at and Lit) : A -* A; a >-* ta are surjective. D

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If / is an algebraic element of a (possibly infinite-dimensional) /¿-algebra A without zero divisors, then the commutative, finite-dimensional subalgebra K[t] is an integral domain. Proposition 3.5.1(b) shows that K[t] is a field. If K[t] is a field, it is usually written in the form K(t), by analogy with (3.1.5). The minimal polynomial of t is then irreducible, or equivalently, prime. Proposition 3.5.2. Suppose K, L, Mare fields, with K <, L <,M. Then dim MK = dim ML ■ dim LK. Proof. If / : / -» L is a basis for LK, and g : / -» M is a basis for ML, then I X J -* M; (i, j) •-» i/ jg is a basis for MK. D Now suppose that the tf-algebra A is an integral domain. Suppose that LK is a finite-dimensional .K-subalgebra of A that is a field. If t is an algebraic element of A over # , then t is also algebraic over L. By Proposition 3.5.2, dim x L(i) = dim L(í)¿ • dim LK, so that Lit) is again a finite-dimensional X-subalgebra of A that is a field. In particular, given algebraic elements tut2,...,t„ of /4 over K, one may define the finite-dimensional iC-field K(tl fr-iXfr). Now the memberships tjl e /£(*!), f, + r2 e £(fj, f2), and txt2 e Ä"(rj, í 2 ) show that the union of {0} and the set of algebraic elements of A form a subfield PAgKA of A. The subfield AlgQC of C is called thefieldof algebraic numbers. Consider a monic polynomial f(T) in K[T]. For any X-algebra A, one may consider f(T) as an element of A[T], since the coefficients of / lie in the subfield K of A. If f(T) = n?^ g /(r - a¡) for elements a, of A, and vl = Xiö!,..., adeg f), then the field ^ is called a splitting field of f(T) in K[T]. For example, the subfield Q(v^) = Q(>/2, - JÏ) of R is a splitting field of T2 - 2 in Q[T], while R is the splitting field of T2 - 2 in U[T]. The existence of a splitting field for f(T) may be proved by induction on the degree of f(T). Suppose that f(T) is a member of the maximal ideal p(T)K[T] of K[T], with p(T) monic. Consider the field L = K[T]/p(T)K[T], a K-algebra. As an element of L[T], the polynomial f(T) has a root a = T + p(T)K[T], so that f(T) = (T - a)fx(T) in L[T] for some element fx(T) of L[T]. Moreover, L = X(a). By induction, / , ( D in L[T] has a splitting field ¿ ( o p . . . , ar). Then / ( D in K[T] has the splitting field K(a, a,,..., a,). The monic polynomial f(T) in X[r] is said to be separable if f(T) = UpjAT - at) with |{a, a d e g / } | = d e g / in any splitting field A = K{au ..., fldeg , ) of f(T) in X[r]. The prime field of a field K is the intersection of all the subfields of K. Thus if K has characteristic 0 (cf. Exercise 3A), the prime field is Q. Otherwise, K has prime characteristic p, in which case the prime field is Z„. Now for a prime p, one has (J + U)p = Zf„Âp\riUp-' = Tp + Up in Zp[T, U]. Thus the map (p : K -* K; a *-> ap is a ring homomorphism of the field K of characteristic p. It is called the Frobenius homomorphism of K. Suppose that K is a finite field, say with prime field Zp. Then the order \K| of K is the prime power q = p", where n is the dimension dim Kz of

COMMUTATIVE ALGEBRA

189

the Zp vector space Kz . Note that K* is an abelian group of order q - 1. For each element a of K, one has cup" = aq = aq~x a = a,so the Frobenius homomorphism

- 1 in K[T] has n roots in K. It follows that n = dx [cf. Exercise 3.2E(a)], so m = 1 and G is cyclic. D For a finite field K of order q, Proposition 3.5.4 shows that the group K* of non-zero elements of K is cyclic. Suppose K* = (e>. Then e is called a primitive element of the field K. There is a group isomorphism (3.5.1)

expe :Zq_l -*K*;x~+

ex.

The inverse loge of (3.5.1) is called the discrete logarithm to base e in K. Note that (3.5.2) for the primitive element e.

K = Zp(e)

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Theorem 3.5.5. / / the fields K and L have the same finite order q = p", then they are isomorphic. Proof. Let K have a primitive element e with (irreducible) minimal polynomial m(T) over Zp. By (3.5.2), there is an isomorphism a : K -» Zp[T]/m{T)Zp[T\. Now I ' - I e m(T)Zp[T], so T" - T = n 6 e L ( r - b) e m(T)L[T]. Thus m(D has a root & in L. Since /n(T) is irreducible in Zp[T], it is the minimal polynomial of b over Zp[T], and there is an injective homomorphism ß : Zp[T]/m(T)Zp[T] -* L with image Zp(b). Since dim[Z p (è)] z = deg m(J) = n = dim L z , the homomorphism 0 surjects, and is thus an isomorphism. Then a/3 : K -* L is the required isomorphism between ÍC and L. D EXERCISES 3.5A. Let t be an algebraic element of a iC-algebra A, with minimal polynomial p(J) = £?igop/>17,,. Suppose that p ( r ) is an irreducible polynomial in K[T]. (a) Show that t is a unit in /I. (b) Show that t~l is algebraic over K. (c) Determine the minimal polynomial of f-1 over K. 3.5B. Determine the minimal polynomials of JÏ, V3~, and -fï + i/T over Q. 3.5C. Determine a splitting field for the polynomial T2 + T + 1 in Z 2 [r]. 3.5D. In the notation of Exercise 3.41, show that Q(wp) is a splitting field for Te - 1 in Q[T]. 3.5E. Suppose that the iC-algebra A is an integral domain. (a) Show that the norm over K yields a monoid homomorphism Alg K ^ -» X. (b) Show that the trace over K yields a /¡C-homomorphism (Alg^^O^ -+KK. 3.5F. Show that the Frobenius homomorphism of the field Zp(T) of rational functions over Z p (cf. Example 3.1.4) is not surjective. 3.5G. Exhibit a non-commutative finite subgroup of the skewfield 0-0 of quaternions. 3.5H. (a) Construct a field K of order 9. (b) Identify a primitive element e of K. (c) Construct a table of the discrete logarithm to base e in K. (d) Decompose K into orbits under the action of the Frobenius automorphism. 3.51. Repeat Exercise 3.5H for a field K of order 8. 3.5J. For a prime p and positive integer n, show that the group GLn(Zp) has an element g of order p" - 1. Show that g has a single invariant factor dx(T) e Zp[T], and that dx(T) is irreducible.

Ill CATEGORIES AND LATTICES

1. POSETS, MONOIDS, AND CATEGORIES There are many interconnections between the concepts of poset and monoid or semigroup. Semilattices as introduced in Section O 4.3 represent an intersection of the two concepts : Semilattices may be regarded simultaneously as posets with greatest lower bounds and as idempotent commutative semigroups. The main topic of this chapter, categories, offers a union of the two concepts. Definition 1.1. A small category C consists of two sets, an object set C0, or ObjC, and a morphism set C I( or MorC. Elements of C0 are called objects or points of C. Elements of Cx are called morphisms or arrows of C. There are two functions assigning points to an arrow : the domain or tail function d0:Cx -» C0; / -» fd0 and the codomain or head function dl:C1 -* C0; f-* fdv For objects x and y of C, the set d^ix) n d^ly) of arrows of C with tail x and head y is denoted by C(x, y). There is an identity function (1.1)

*:C0-»C,;*~1X

with ed0 = ed1 = \c¡¡, so that Vx e C0, lx e C(x, x). Finally, for each triple x, y, z of objects of C, there is a composition (1.2)

C{x,y) X C{y,z) ^ C(x,z); (f,g) ~ fg

such that (1.3)

Vx,yeC 0 ) V/eC(x,y),l x / = / = / l y 191

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and (1.4) V*, y.z.te

C0> V/ e C(x, y), Vg e C( V, z), VA e C(z, r),

(£)*-/(«*)• One usually refers to (1.4) as the associative law of composition.

□

Note that the identity function e of (1.1) injects, since it has the head and tail functions as retractions. One could thus identify C0 with CQe, and suppress explicit mention of the objects altogether. An element / of C(x, y) may be represented pictorially as x-*y / : x -+ y. In this convention, the composition (1.2) appears as x

/

► y

g

or

► z

(1.5)

Example 1.2 (Monoids). Let (M, •, 1) be a monoid. Define M0 : = {1} and Ml: = M. Define the tail function dQ : M -* {1} and the head function dx:M -» {1}. Thus M = C(l, 1). Define the composition C(l, 1) X C(l, 1) -> C(l, 1); (m, n) -» mn by monoid multiplication. Define e : {1} -> M; 1 -> 1. Then (1.3) and (1.4) hold by (O 4.3) and (O 4.2). Thus M forms a small category, with a single object. Conversely, composition in a small category with a single object forms a monoid (Exercise 1A). D Example 13 (Posets). Let (X, a) be a poset. Define a0 : = X and a1 : = a. Define the tail function d0 : a -» X; {x, y) •-» x and the head function d1: a -* X; (x, y) -* y. Using the reflexivity of a, define the identity function to be the diagonal A : X -* a; x <-* (x, x). Thus VJC, y e X, a(x, y) = if xay then {(x, y)} else 0 . A composition (1.2) is then defined (uniquely) by the transitivity of a, and a becomes a small category. D Example 1.4 (Matrices). Let 5 be a unital ring. Let 5 0 be the set of positive integers, and define S1: = Ui ¿ />m S¡" to be the set of all matrices over 5. Define d0 as the disjoint union of the maps 5/" —* {/} and ¿fj as the disjoint union of the 5,m -» \m). Define e:m<-*Im. Note that S(l, m) = 5;m for 1 <.l, m. Define the composition S¡" X 5 J - » S" by matrix multiplication. Then S becomes a small category. D Example 1.5 (Systems of Sets). Let C0 be a set of sets. For X,Y & C0 let C(X,Y) = Yx, the set of all functions from X to Y. Let Cl = U* Yec C(X, Y). Define d0 : Ct -> C0; / -> Dom / and dx : Cx -► C0; / -> Cod /

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193

Define e : C0 -» Cjj X >-* id*. Define the composition (1.2) by the usual composition of functions. Then C becomes a category. D Example 1.6 (Systems of Quasigroups). (a) Let C0 be a set of quasigroups. For X,Y& C0, let C(X,Y) be the set of all quasigroup homomorphisms from X to Y. Make the remaining definitions as in Example 1.5. Then C becomes a category, since the composition of quasigroup homomorphisms is a quasigroup homomorphism. This example may be mimicked using other structures on the sets X, Y in place of quasigroup structure, taking C(X, Y) to be the set of homomorphisms of the structure, i.e. the set functions / : X -*Y preserving the structure. (b) Again taking C0 to be a set of quasigroups, one may define C(X, Y) to be the set of all quasigroup homotopies from X to Y. Define «/i. / 2 . / 3 ) : (*o, ■ , / , \ ) - (*i. • - A\ M : = X-t for / = 0, 1, and e : (X, ■, / , \ ) •-» (1*, lj-, 1*). Define the composition (1.2) by the composition of homotopies (cf. Exercise I 4A). Then C becomes a category. D Suppose that a small category C is given as in Definition 1.1. One may then define a new category Cop as follows. Take C£p = C0 and C,op = Cv Take d%p = d, and d°p = d0. Thus the arrows of C are reversed (turned head to tail) to yield the arrows of Cop. Moreover, Cop(x, y) = C(y, x) for x,y eCQ = C£p. Take eop = e. The composition C°t>(x,y) X C°v(y,z)

-

C°P(*,Z);

(f,g)

^/»g

is defined in terms of the composition in C as C(y, x) X C(z, y) ^ C(z, * ) ; (/, g) - «f. For example, the diagram (1.5) is replaced by / x *—-— y *

g

z

(1.6) X *

■z

f°g

Z

The category Cop is called the opposite or dual of the category C. For example, if M = (M, •, 1) is a monoid, determining a category M as in Example 1.2, the dual of the category M is the category determined by the opposite monoid M op = (M, °,1) as in (I 1.4). If C is the category determined by the power set poset (2B, ç ) of a set B according to the construction of Example 1.3, then C op is the category determined by the poset (2B, 2). In general, for a poset (X, a), the poset (X, a') with xa'y <=>_ya;c is called the

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dual of the poset (X, a). For any category C, the dual C opop of the dual C op is just the original category C. A directed graph V = (r 0 , T,, dQ, dx) is a pair (d0 : T, -* T0, dx : Tx -* T0) of maps. For example, suppose that a is a simple directed graph on a vertex set X (in the sense of 1.3), i.e. an antireflexive binary relation on X. Then d0 : a -* X; (x, v) -» x and dx : a -* X; (x, y) ^ y yield a directed graph (X, a, d0, dx) in the current sense. A non-trivial path in the directed graph T is an element fx--.fn of the free semigroup Yx over Tx with the property VI <, i < n, fidl = fi+1âQ. One may then construct a category m , the path category of the graph T, as follows. Let m , be the disjoint union of the set r 0 and the set m * of non-trivial paths in I\ In this context, elements of TQ are called trivial paths, and elements of r r i j are called paths in I\ Let r n 0 = r 0 . Define d0 : Til, -* r i l 0 to be the disjoint union of the identity T0 -» r 0 and the map rTIj1" -» T0; / , . . . /„ -> fxd0. Similarly, define d, : m , -» m 0 as the disjoint union of the identity r o -» T0 and m + -» T0; / , . . . / „ >-*/„
»{0}

{1}

>{0,1}

The edges in Hasse diagrams are often drawn without arrowheads, using the convention that the tail of an edge is "lower than" or "south of' its

POSETS, MONOIDS, AND CATEGORIES

195

head. Using this convention, the Hasse diagram of (2{
"^{1} •

0 (Advocates of this convention are wont to claim that the Hasse diagram of the dual of a poset is the Australian version of the Hasse diagram of the original poset.) D Example 1.8 (The Nerve of a Category). A small category C, as in Definition 1.1, yields a directed graph C = (C0,C1,dQ,dl). The path category CTI is called the nerve of the category C. Elements of the intersection C„ : = Cüj n C" of the set of paths with the uniform code of length n over Cl are known as n-simplices in C. For example, the simplices in (the category determined by) the (dual of the) poset of ideals of a principal ¡deal domain A are just the simplices classifying finitely generated /1-modules via the rational canonical form. The successive points along such a path are the invariant factors of the module. D EXERCISES 1A. Verify that if a small category C has a single object, then C1 forms a monoid under composition. IB. (a) Generalize Example 1.3 to show that each pre-ordering yields a small category, (b) Suppose that for each pair (x, y) of objects of a small category C, the set C(x, y) has at most one element. Show that Cj is a pre-ordering on C0. IC. For the system C0 = {0,{O},{0,1}} of sets, determine the corresponding category C given as in Example 1.5. [Hint : Represent C by a picture of points and arrows, as in (1.5).] ID. Construe the dual of the category S of Example 1.4 in terms of transposition of matrices. IE. Show that the free monoid A* over an alphabet A is obtained (according to Exercise 1A) as the path category of the directed graph ({1}, A, A -» {1}, A - {1}). IF. (a) Show that the poset (IdZ, ç ) of ideals of the ring of integers is not locally finite, (b) Show that the poset ((IdZ) - {0}, ç ) of non-trivial ideals of the ring of integers is infinite, but locally finite.

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IG. Let (X,a) be a locally finite poset. Give a formal proof that a is specified uniquely by its Hasse diagram. 1H. A poset (X, <,) satisfies the ascending chain condition if xt <, x2 ^ ••• ^ xn <, ■•■ => 3m. xm =xm+1 *=xm+2 = •• . Show that the ascending chain condition does not imply local finiteness. II. Consider the problem of storing a partial order in a computer. Discuss the relative efficiencies of the two following techniques : (a) Hasse diagrams; (b) intersections of total orders [cf. Proposition O 3.5.4(b) and Exercise O 3.5E]. 1J. Let (X,a) be a locally finite poset. Let K be a unital, commutative ring. Then the set Ka of functions a -» K is a ^-algebra (Ka,-,C) with pointwise operations. In particular, the identity Ç is the constant (or zeta) function f : a -» {1}. Now define a new operation of convolution on Ka by f*g{x,y)=

E ts[x,y)a

f(x,t)g(t,y).

(a) Show that (Ka, *, S) is a X-algebra with pointwise K-module structure and identity S(x, v) = if x = y then 1 else 0. (b) If K has characteristic 0, prove

W ( * , J 0 - \XU(x,y)

na2\,

where a2 is the uniform code of length 2 over a (cf. Exercise O 4.1B). Similarly, express \Xll(x, y) O a"|, for all positive integers n, in terms of the zeta function. (c) Show that £ is invertible in (Ka,*,£). Its inverse ¡i is called the Möbius function of(X,a). (d) Show that 8 is invertible in (Ka,-,() if (X,a) is an antichain, i.e. a = X. (e) Let (X, y) be a chain [cf. (O 3.5.2)]. Prove fi(x, y) = if x = y then 1 else if |U, y]J = 2 then - 1 else 0. 1.1. Diagonalization and Large Categories Example 1.5 made a category C with a given set of sets as the set C0 of objects of C, and with C(x, y) as the set of functions / : x -* y for sets x,y in C0. One would like to be able to consider all sets together in such a category. However, the class of all sets does not itself form a set. To see this, consider the property of sets given by the following diagonalization argument due to Cantor.

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Proposition 1.1.1. There is no surjection f : A -* 2A from a set A to its power set. Proof. Suppose that there were such a surjection /. For the subset B = {JC e A \x € xf), there would then be an element a of A with B = af. This would lead to a contradiction, for a e B => a £ af = B, while a i f i = » aeaf=B. D Corollary 1.1.2. The class of all sets does not form a set. Proof. Suppose that the class of all sets were to form a set A. Then the power set 2A, as the set of subsets of A, would be a subset of A. Let j:2A -»A; X -» X be the inclusion. As an injection, the function j would have a right inverse / with jf = 12A (Proposition O 3.1.1). Then / : A -* 2A, having ;' as a left inverse, would surject (Exercise O 3.1C). This would violate Proposition 1.1.1. □ Corollary 1.1.2 shows that the class of all sets is a concept located at a higher hierarchical level than that of sets. Each set is a class, and one may characterize sets as those classes that are members of a class (e.g. the class of all sets). The remaining classes, such as the class of all sets, are called proper classes. One may operate with classes and functions between them, just as one operates with sets and functions between them. If the sets appearing in Definition 1.1 are replaced with classes, then one obtains the definition of a large category. For example, the category Set of sets has the class Setn of all sets as its class of objects. For two sets X, Y, the class Set {X, Y) is the set Yx of functions from X to Y, and the class Set, of arrows of Set is the disjoint union L^,y e sju Set(Ar, Y). Note that this union of sets is a proper class, since it contains the class Setn of all sets as a subclass, namely as the image of the identity function e : Set0 -» Set,; X -* iáx. The tail function in Set is d0 : Set, -» Setn; / - > D o m / , while the head function is d, : Set, -» Set0; / -* Cod/. The composition in Set is the usual composition of~set functions. A large category C is said to be locally small if C(x, v) is a set for each pair (x, y) of objects of C. Note that Set is locally small, since Set(-Y, Y) is the set Yx. Example 1.1.3 (Categories of Quasigroups). Just as the category Set is a large category corresponding to the small categories of Example 1.5, so~there are large categories corresponding to the small categories of Example 1.6. (a) The (large) category Q of quasigroups has Q,, as the class of quasigroups. For quasigroups P and Q, the class QjiP, Q) is the set of quasigroup homomorphisms from P to Q. Then Q, = \JPQ e o Q(P, Q). The tail and head functions are given by the domain and codomain respectively, while the composition is the usual composition of functions.

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(b) The homotopy category Qtp of quasigroups again has Qtpn as the class Q^ of quasigroups. For quasigroups P and Q, the class Qtp(P, Q) is the set of quasigroup homotopies from P to Q. Then Qtpn = UP p e ^ Qtp(F, Q). Other details are as in Exercise 1.6(b). D Example 1.1.4 (A Large Semilattice). The large poset or poclass (Setn. ç ) yields a large category by the construction of Example 1.3. The corresponding large semilattice (Setfl,U ) yields a large monoid (Set n ,u,0), which in turn yields a large category by the construction of Example 1.2. Note that this large monoid M has a singleton set {1} as M0, while M(l, 1) = Ml is the proper class of sets. In particular, M is not locally small. D Example 1.1.5 (Partial Functions). The large category Pfn has the class Set0 of sets as Pfnn. Given sets X and Y, a partial junction f from A' to Y is a JJotal or "ordinary") function f:Xf->Y from a subset Xf of X to Y. The subset Xf of X is called the domain of definition Dod / of /. Then PtnjX, Y) is the set of partial functions from X to Y, and Pfiij is the class U y v . y , ?fn(X,Y). The composite of two partial functions / : X -* Y and g:Y-* Z is the partial function fg:X-*Z whose domain of definition is (f)~1(XffnYg), and such that Jg is the total function composite of the restriction of / to this domain with the restriction of g to Xff n Yg. For a set X, the identity 1^ is the total identity function id^, construed as a partial function with domain of definition X. The category Pfn is called the category of partial functions. O EXERCISES LIA. Define £ :2 N -> [0,1]; A -» 2I^,.03-n-lXA(n). (a) Show that f is a well-defined injection. (b) A set X is countable if there is a surjection N -* X. Using (a) and Proposition 1.1.1, show that the closed real interval [0,1] is uncountable. LIB. (a) Show that the abelian group Li „ e N Z is countable. (b) Show that the abelian group n „ e N Z is uncountable. 1.1C. Define a set S of functions / : M -» N inductively by the following rules: (a) V/i G IM, (IM -» M) e S; (b) 1N e S; (c) V/, geS,(n~nf + ng) G 5; (d) V / , g G S, (n ~ n/.Hg) G 5;

(e) V/, g e 5, (n - (n/)"«) e 5. Show that there is a function / : N -» M f which is not a member of S.

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LID. Exhibit large categories C with each of the following two properties: (a) C0 is a set and Cx is a proper class; (b) both C0 and Cx are proper classes. LIE. Set up a large category (Mon; Set) with (Mon; Set)» as the class of monoid actions and with (Mon; SetXiA, M), (ff, NJ), as the set of action morphisms from the M-set A to the TV-set ff (cf. Section I 1.2). Define the tail, head, composition, etc. 1.1F. Set up a large category Rel of relations, with Rel» = Setn, as follows. For sets X and Y, one has Reli*, Y) = 2XxY. For a e RcViX, Y) and ß G Rel(y, Z), define the composition of a and ß to be the relation product a ° ß = {(je, 2) 6 Z X Z|3.y e Y. jcay/3z}. Verify that Rel is a category. 1.1G. Use the usual constructions of set theory to show that, for sets X and Y, the class Pfn(A", Y) is a set. 1.1H. Show that a rational function p(T)/q(T) in the field of fractions R(r) of U[T] yields an element r <-* p(r)/q(r) of PfaflR, R) with domain of definition {JC e R|Ç(JC) # 0}. Can you put an R-algebra structure on PfnflR, R) so that the R-algebra homomorphism U[T] -» Set(R, R) of ÜT3.5) extends to an R-algebra homomorphism R(D -» Pfa(R, R)? [Hint : Try Dod(/ - g) = Dod / n Dod g in Pfn(R.R).1 1.11. The graph of a partial function / : X -* Y is the subset gr/={(x,>0e(Dod/)xY|3>=*/} of A!" x Y Define a relation ^ of extension on Pfn(AT, Y) by / ^ g iff g r / C g r f • Show that (Pfn(Ar, Y), ¡s) is a poset. l.lj. Consider the following program to compute a natural number nf for each natural number n : begin Í : = 0, 0/ = 1 while 1 < n do (/ + 1)/: =if.(i + 1) i:=i +1 endwhile output nf (a) Show that each run of the program builds a partial function fn:N -* N whose domain of definition is the closed interval [0, n] in (IM, 2;). (b) Show that {/„In e N} is a chain in (PfnflUH).<.). What is the upper bound of the chain?

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1.2. Functors and Concrete Categories A monoid homomorphism / : (M, •, 1) -* (N, •, 1) is a function f:M-*N with 1/= 1 and with yimx,m2 G M,(mlm2)f = (mJXm2f). For example, the exponential function is a monoid homomorphism exp : (R, +, 0) -* ((0, °°), •, 1) from the reals to the positive reals. An order-preserving or monotone {increasing) map f:(X,a) -* (7,0) between posets is a function f:X-*Y such that (*„ JE2) G a =» (*i/>^2/) e £• For example, the exponential function is a monotone map exp : (R, <,) -» ((0,°c), ^ ) from the reals to the positive reals. The concept of a category gives a common generalization of the concepts of monoid and poset. There is a corresponding common generalization of monoid homomorphisms and monotone maps : the concept of a functor. Definition 1.2.1. Let C and D be categories. Then a (couariant) junctor F : C -» D consists of two functions, an object part F :C0 -» D0 and a morphism part F :Cl -» D„ with the following properties : (a) V*, y G C0, CU;, y)F ç D(xF, yF); (b) Vx G C0, 1XF = l , f ; (c) Vx,y,z G C0, V/ e C(x, y), Vg G C(y, 2), (/g) F = / F • gF. D Example 1.2.2. Let Y be a fixed set. Given a set Z, define ZFY : = Z XY. Given a function / : Z, -* Z 2 , define / F y :ZyFY-» Z2FY to be the function (/, l y ) : Z, X Y ^ Z 2 X Y; (z, y) -» (z/, y) given by (O 3.2.4). Then Fy : Set -» Set is a functor. D Example 1.23. Let (X,a) be a poset, construed as a small category a according to Example 1.3. For * G AT, define the down-set xD = {y G X\(y, x) G o}. Now for (x p x 2 ) ^ a, there is a unique arrow / : xx -* x2 in the category a. Define fD : xxD -* x2D to be the inclusion function of the subset xxD in the set x2D. Then D : a -* Set is a functor. O Example 1.2.4. Let C be a category, and let CXI be its nerve (Example 1.8). (If C is large, then so is CTI.) There is a functor £c : CU -» C whose object part is the identity on C0 = CU0 and whose morphism part maps a non-trivial path in C to the composite of its constituent arrows. D Example 1.2.5. Let C be the (category determined by) the poset with Hasse diagram (0,0) - ( 1 , 0 )

(1,1) - . ( 2 , 1 ) .

Let D be the (category determined by) the poset with Hasse diagram 0 -» 1 -* 2. Let F be the monotone function F : C -» D; (x, y) *-* x. Then F is a functor F :C — D. Note that the object image set C0F = {0,1,2} and the

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morphism image set C,F = {1 0 ,1,, 1 2 , 0 -» 1, 1 -» 2} do not form an "image category," since there is no composite of 0 -* 1 and 1 -» 2 in CXF. D Example 1.2.6. A functor P : Set-» Setop, the contravariant power set functor, is defined by F : Setß ~* S e J p ; T -* 2X and P:Set, -> Set°p; ( / : X -► Y) (/"' : 2 y -» 2*), as in Exercise O 3.4E. One may also construe P as a functor P ; Setop -> Set. D In general, for categories C and D, a functor F :C -* Z)op, or equivalently F : Cop -» Z), is sometimes described as a contravariant functor F :C -» D from C to Z). If C and Z) are poset categories, then a contravariant functor F : C -» Z) is called an antitone or order-reversing map from C to Z). If G is a group construed as a monoid category, then inversion J:G -» G; g -> g - 1 yields a contravariant functor from G to itself. Given functors F :C -* D and G : Z> -» E, the respective composites FG of the object and morphism parts of F and G determine a composite functor FG :C-*E. Each category C has an identity functor \C:C-*C whose object and morphism parts are the respective identity functions l c and l c . One may thus form a category Cat of small categories, with Catn as the proper class of small categories. Given small categories C and D, the class Cat(C, D) is the set (Dc)0 of functors from C to D. Composition in Cat is defined by composition of functors. For arbitrary categories C and D, a functor F :C ^> D is an isomorphism if there is a functor G:D-*C with FG - l c and GF = lD. In this context, the categories C and D are isomorphic, a relation denoted by C = D. The functor G is the {two-sided) inverse F _ 1 of F. A functor F :C -* D is ./u// if, for each pair A", Y of objects of C, the function C(X, Y) -* DÍAT, YF); / -> /F surjects. For example, let Gp_ be the large category of groups and homomorphisms between them. Th~en the inclusions Gpn «-» Qp and Gj), •-» Q, yield a full functor from Gp to the large category Q of quasigroups introduced in Example 1.1.3(a). In general, a category C is said to be a subcategory of a category Z) if there are inclusions C0 ^ D 0 and C, «-» D1 yielding a functor C «-» D, the inclusion functor. For example, G£ is a subcategory of Q and Q is a subcategory of Qtp [via Ql -» Qt£,; {p.P-*Q)** ((/, f,f):P-* Q)]. The subcategory C ÔÏD is said to btfull if the inclusion functor C *-» Z) is full. Thus G_£ is a full subcategory of Q, but Q is not a full subcategory of Qtp. Note thalTonce a category D is given, a full subcategory C of Z) is specified completely by giving the object subclass C0 of Z)0. A functor F : C -» Z> is faithful if, for each pair X, Y of objects of C, the function CÍA', Y) -» D(XF, YF); f-* fF injects. For example, the inclusion functor of any subcategory is faithful. As a second example, consider a module X over a unital ring 5. Construe the monoid (5,-,l) as a category S with 5 0 = {1} and 5, = 5, as in Example 1.2. Construe the representation (II 2.1) as a functor R : S -* Set, with object part Z? : 1 -» Z and morphism

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part R : s -» (x -» xs). Then A" is a faithful S-module iff the functor R is faithful. A concrete category is a category C supporting a faithful functor C -» Set. the underlying set Junctor. In Section O 3, algebra was characterized as the "study of sets with structure." In studying sets with structure, algebra also studies functions preserving that structure. Since the identity map on a set preserves its structure, and since the composite of structure-preserving maps preserves the structure, algebra is a rich source of concrete categories, such as Q and Gp, of sets with structure. The underlying set functor from such a category tEeñ just forgets the structure of an object, "remembering" only the underlying set. Its morphism part forgets the structure-preserving property of a function, "remembering"only the function itself. Functors forgetting (some) structure in this sense are known as forgetful functors. For example, the underlying set functor i/:Q-> Set has object part U:(Q,-,/,\) -» Q and morphism part U : ( / : (/»,-, / , \ F ^ (Ö, ■. / , \ )) -» ( / : P -» Q). Example 1.2.7. Here is a catalogue of concrete categories of "sets-with-structure" and "structure-preserving maps" that have arisen so far. Q—Quasigroups and quasigroup homomorphisms. Mon—Monoids and monoid homomorphisms. Gp—Groups, a full subcategory of Q and of Mon. Set—Sets and functions (no structure). Sgp—Semigroups and semigroup homomorphisms. SI—Semilattices, a full subcategory of Sgp. M (For a fixed monoid M)—Msets and Ai-homomorphisms. Lp—Loops and loop homomorphisms. RQ—Right quasigroups and right quasigroup homomorphisms. RLp—Right loops and right loop homomorphisms. Rng—Non-unital rings and non-unital ring homomorphisms. Ring—Unital rings (rings with "i-dentity element") and unital ring homomorphisms. Modj (For a fixed unital ring S)—Unital right S-modules and S-homomorphisms. 5 Mod (For a fixed unital ring S)—Unital left S-modules and S-homomorphisms. Alg^ (For a fixed unital, commutative ring K)—iC-algebras and /C-algebra homomorphisms. D

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The concrete categories of Example 1.2.7 all have obvious underlying set functors, namely the forgetful functors. In other concrete categories, the underlying set functors may not be quite so obvious. Example 1.2.8. The category Qtp of quasigroups and homotopies is a concrete category. The faithful underlying set functor I/;Qtp-» Set has object part U:Q-*QVQUQ and morphism part U:(fvjÇJ3) - » / , U / 2 U/ 3 . Note that the composite of the inclusion functor Q^ Qtp with U : Qtp-» Set gives an underlying set functor for the concrete category Q that is different from the forgetful functor. D Example 1.2.8 shows that a concrete category, such as the concrete category of quasigroups, may have several different underlying set functors. Thus it is sometimes necessary to consider a concrete category C along with a specified underlying set functor U:C -» Set. Following the category theory tradition of flippant definitions that goes right back to the beginning of the subject, one might define such a pair (C, U) as a reinforced concrete category. Example 1.2.9. The category Pfn of partial functions (Example 1.1.5) is a concrete category, with an underlying set functor G : Pfn-» Set known as the {one-point) compactification functor or Grimeisen operator. Fix a singleton {*}. (In computer science, one may choose to denote * as the element ± , "bottom"or "undefined." In mathematics, one may choose to denote * as the element °o, "the point at infinity.") Then G has object part G : X <-* X U {*} and morphism part G : ( / : X -» Y) -* ( / U ((X - Xf) -> {*})U (1{.}). □

EXERCISES 1.2A. Verify that F:Set-» Set in Example 1.2.2 really is a functor. 1.2B. (a) Show that the composite of monotone maps between posets is again monotone. Conclude that there is a large category Pos with Posn the class of ordered sets and Pos, the class of monotone functions between them. (b) Show that Pos is a full subcategory of the large category Cat, via the construction of Example 1.3. (c) Construing each set X as the antichain or discrete poset (X, X), show that Set is a full subcategory of Pos and Cat. (Cf. Exercise O 4.3A.) Sets regarded in this way are known as discrete categories. 1.2C. Let F : C -> D be a functor. (a) If C is a poset, show that F is faithful. (b) If D is a poset, is F necessarily full?

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1.2D. Give an example of a full and faithful functor that is not an isomorphism. 1.2E. Is Gp_ a full subcategory of Qtp? 1.2F. Let F : C -» D be a full functor. Show that the image of F, with object class CQF and morphism class C,F, forms a category. (Cf. Example 1.2.5.) 1.2G. Show that there is a covariant power set functor PiSet-» Set with P : Seto -» Set„; * - » 2* and P : Set, -> Set,; ( / : X -» Y) -> (2* -* 2Y; A ~* Af). 1.2H. Let G be a fixed abelian group, and let 5 be a unital ring. Show that duality as in Section II 2.1 yields a contravariant functor T : Modj -> s Mod with object part Xs -*SXT = Hom(Ar,G) and morphism p a r t T ^ / r = Hom(/,G). 1.21. (a) Show that 0 1

(12)

1 0

and (23)

1 1

0 1

extend to (the morphism part of) a functor 5 3 -» Z 2 from the symmetric group 5 3 (construed as a monoid category according to Example 1.2) to the unital ring Z2 (construed as a matrix category according to Example 1.4). (b) Show that

(12)

0 1 0

1 0 0

0' 0 1

and (23)

1 0 0

0 0 1

0] 1 0

extend to a functor 5 3 -» Z2. 1.2J. A forgetful functor need not be completely forgetful. For example, there is a forgetful functor U :Mon-* Sgp with object part U : (M, ■, 1) -> (M,- ), just forgetting the existence of the identity element. Excluding Set from the catalogue of Example 1.2.7, determine all the remaining forgetful functors U:C-*D connecting distinct categories C,D from the catalogue. 1.2K. Verify that U in Example 1.2.8 is a faithful functor. 1.2L. Verify that G in Example 1.2.9 is a faithful functor. 1.2M. Show that the category M of Example 1.1.4 is not concrete. 1.2N. Fix a unital ring 5 and a positive integer n. Define U : Modt -» Set by t/:Mod,) n -»Set ft ; V~ V and U:Mod.\ -> Set,; ffT^-> Wl~ ( ( / , . . . , / ) : V -* Wn). Show that U is a faithful functor. 1.20. Show that each small category is concrete.

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1.2P. Let n be a natural number, and let K be a commutative unital ring. (a) Show that there is a contravariant functor K"( ;JO:Mod y -» Mod,, with object part V -» K"(V;K) and morphism part 9 •-» K"(6;K). (b) Show that there is a contravariant functor Kn( ;K):Modr -* Modjç with object part V -* K"(V; K) and morphism part 6 -»

Kn(b7K).

1.2Q. Show that the function Gpn -» Setn; G -> Z(G) assigning the center to a group cannot be the object part of a functor Gj>-> Set. 1.2R. For a set X, let ZF be the set of equivalence relations on X, ordered by inclusion and considered as a poset category. For a function / : X -» Y, define fF :YF -+XF; a -* ker ( / nat a). Show that F is a functor F : Setop -» Cat. 1.3. Commuting Diagrams, Epimorphisms, and Monomorphisms One of the guiding themes in the study and use of categories is the expression of mathematical statements pictorially, using diagrams of points and arrows (category theory as "comic book mathematics") Such an expression tends to lead to a better understanding of the mathematical statement involved. It also facilitates the transfer of mathematical content from one context to another—specifically, from one category to another. This is a valuable tool for overcoming the progressive fragmentation and specialization that bedevils mathematics (along with other scientific disciplines). Formally, a diagram in a category C is a functor D : TU -* C from the path category of a directed graph T to C. (In a picture of D, one usually just displays r¡D oí T0D U rlD.) The diagram is said to commute if, for any two non-trivial paths /, g in T with common heads and tails, the images fD and gD in C agree. For example, (O 3.2.4)-(0 3.2.6) are commuting diagrams in Set. Consider the following simple mathematical statement : (1.3.1)

The function e : X -» Y surjects.

To express (1.3.1) pictorially, one may use the dual of Proposition O 3.1.1 (cf. Exercise O 3.1C) stating that (1.3.1) is equivalent to V/ : Y -> Z, Vg : Y -> Z, ef = eg => / = g. Thus the commuting of each diagram

(1.3.2)

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in Set implies / = g. If e : X -» Y is a morphism in a category C such that the commuting of each diagram (1.3.2) implies / = g, then e is said to be an epimorphism in the category C. In particular, epimorphisms in the category Set are just epimorphisms in the sense of (O 3.1.5). Example 1.3.1. Consider the category Ring of unital rings (cf. Example 1.2.7). If a ring homomorphism e : R -* S is surjective, then e is an epimorphism in Ring. However, the ring homomorphism e : Z -» Q embedding the integers in the rationals is an epimorphism in Ring that does not surject. D A morphism r : X -* Y in a category C is said to be left invertible or a retraction if there is a morphism s : Y -» X such that (1.3.3)

sr=l Y-

The dual of Proposition O 3.1.1 shows that the concepts of retraction and epimorphism are equivalent in the category Set. On the other hand, each morphism in a poset category (Example 1.3) is an epimorphism, while only the identities are left invertible there. In a general category C, all that survives of the dual of Proposition O 3.1.1 is the observation that retractions are epimorphisms (Exercise 1.3E). To obtain the dual of a concept in a category C, one may interpret the concept in the dual category Cop. The dual of the concept of "epimorphism" is the concept of "monomorphism."Thus a monomorphism m : Y -* X in a category C is a morphism such that the commuting of each diagram

(1.3.4)

[the dual of (1.3.2)] in C implies f = g. Note that monomorphisms in the category Set are just monomorphisms in the sense of (O 3.1.2). A morphism s: Y -» X in a category C is said to be right invertible or a section if there is a morphism r : X -» Y such that (1.3.3) holds. Applying the result that "retractions are epimorphisms" (Exercise 1.3E) to the category C op , one obtains the result that sections are monomorphisms in any category C. The relationship between these two results is described as duality: thus "retractions are epimorphisms" is dual to "sections are monomorphisms." In (1.3.3), the morphism s is called a left inverse of the left invertible morphism r, and r is called a right inverse of the right invertible morphism s. A morphism in a category is invertible or an isomorphism if it has both a left inverse and a right inverse, i.e. if it is both a retraction and a section.

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Proposition 13.2. Let f be a morphism in a category C. Then f is an isomorphism iff it has a unique left inverse and a unique right inverse that coincide. Proof. The " i f part is immediate. Conversely, suppose that / : X -* Y is an isomorphism, with a left inverse s and a right inverse r. Then s = s\x = sfr = \Yr = r, so that s and r coincide. Moreover, if s' is also a left inverse, then s' = r = s. Similarly, the right inverses are unique. D The coincident right and left inverses of an isomorphism / are denoted by f~l, the inverse of the isomorphism /. A pair X, Y of objects of a category C are said to be isomorphic, written X = Y, if they are the domain and codomain of an isomorphism / : X -* Y in C. For the concrete categories of Example 1.2.7, and for the category Cat of small categories, this concept of isomorphism agrees with earlier ones. Note that quasigroups P and Q are isomorphic in Qtp iff they are isotopic. If M is a monoid realized as a category according to Example 1.2, then the isomorphisms in the category M are the units of the monoid M. Mathematical statements formulated in set-theoretical terms generally involve elements. In order to translate such statements into pictures of points and arrows, one often needs a pictorial way of representing elements. Now an element x of a set X is the unique element of the image of a function {°°} -» X; oo -> x from a singleton to X. Singleton sets are characterized by the property that there is a unique function from any given set to the singleton. The concept of a terminal object in a category abstracts this property of singletons in Set. Definition 1.3.3. (a) An object T of a category C is terminal if C(X, T) is a singleton for each object X of C. (b) An object _L of a category C is initial if C(±,X) is a singleton for each object X of C. (c) An object 0 of a category C is a zero object if it is both initial and terminal. □ Note that initial objects in a category C are terminal objects in the dual category C op : The concept of initial object is dual to the concept of terminal object. The empty set is initial in the category Set. An upper bound is terminal in a poset category. The zero group {0} is a zero object in the category Mod7. A unital, commutative ring K is initial in the category Algy. A category may have no initial or terminal objects (Exercise 1.3J). In Set, each singleton is a terminal object, and any two singletons are isomorphic. This behavior is typical. Proposition 13.4. Any two terminal objects of a category C are isomorphic.

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Proof. Suppose that C has terminal objects T¡ and T2. Consider the diagram Tx

* T2

> T2

r,

» r,

> T2

in which each arrow is uniquely specified since its codomain is terminal. The diagram commutes, since each path in the diagram has a terminal codomain. Commutativity of the left square shows that T2 -» Tl is left invertible. Commutativity of the right square shows that T2 -» Tx is right invertible. Thus T2 -> 7, is an isomorphism. D Corollary 1.3.5. (a) Any two initial objects of a category C are isomorphic. (b) Any two zero objects of a category C are isomorphic. Proof, (a) The dual of Proposition 1.3.4, and (b) A special case of Proposition 1.3.4. o The apparent triviality of Corollary 1.3.5(a) is deceptive. In fact, the result is the basis for a major theme of category theory: the specification of mathematical constructs as initial objects in categories. For example, the ring of integers is specified [by Corollary 1.3.5(a), to within isomorphism] as an initial object in the category of unital commutative rings. One may view this as a precise formulation of the vague idea that "you can calculate with commutative rings the same way you calculate with integers." EXERCISES 1.3A. Show that a small category C is a pre-ordering (cf. Exercise IB) if the functor BC of Example 1.2.4 is a commuting diagram. 1.3B. Interpret each of (O 3.2.4)-(0 3.2.6) formally as a commuting diagram in the category Set. (For each, exhibit a suitable directed graph T and functor D : m -> Set.) 1.3C. Verify the content of Example 1.3.1. 1.3D. Show that a composite of epimorphisms in a category C is again an epimorphism in C. 1.3E. In a category C, prove that left invertible morphisms are epimorphisms. 1.3F.

;

p

Let 0 - > / ! - > £ - > ô - > 0 b e a n exact sequence of abelian groups. (a) Show that p is an epimorphism in the category Mod7 (cf. Example 1.2.7).

POSETS, MONOIDS, AND CATEGORIES

1.3G. 13H. 131.

1.3J. UK. 1JL.

1JM.

209

(b) Show that p is a retraction iff the exact sequence splits (cf. Exercise II 1.1D). (c) Show that ; is a monomorphism in the category ModT. (d) Show that ;' is a section iff the exact sequence splits. Formulate and do the dual of Exercise 1.3D. Show that isomorphism in a category C yields an equivalence relation on C0. Show that the following three properties of a morphism / of a category C are equivalent : (a) / is an isomorphism; (b) / is a monomorphism and a retraction; (c) / is an epimorphism and a section. Let Twoup be the full subcategory of the category Set comprising sets with at least two elements. Show that Twoup has no terminal object and no initial object. Show that the empty set is a zero object in the category Pfn of partial functions. Determine initial objects in each of the concrete categories of Example 1.2.7. In particular, contrast the initial objects of the categories Rng and Ring. Let Fjd be the full subcategory of Ring whose objects are fields. Show that Fjd has no initial object and no terminal object.

1.4. Natural Transformations and Functor Categories Category theory studies three levels of transformations. At the first level, categories themselves comprise morphisms between objects. At the second level, functors operate between categories. At the third level, natural transformations operate between functors. Definition 1.4.1. Let S :C -* D and T :C -» D be functors with common domain and codomain. Then a natural transformation T : 5 -> T between 5 and T is a function (1.4.1)

T-.CO^D^X^ÍTX-.XS^XT)

such that, for each morphism X -* Y of C, the diagram XS — * - » XT (1.4.2)

fS

fT

TS ——► YT

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commutes. The D-morphism rx : XS -* XT is called the component of r at the object X of C. The natural transformation is called a natural isomorphism if each of its components is an isomorphism of D. D Note that, to specify a natural transformation, one just has to specify its components, in such a way that the diagrams (1.4.2) commute. Example 1.4.2. Let M be a monoid, construed as a category M with M0 = {1} and Mx= M according to Example 1.2. Let (A, M) be an M-set via representation 5 : M -» AA and let (B, M) be an M-set via representation T:M-+BB. Construe 5 as a functor 5 : M -» Set with object part S : 1 -» A and morphism part S : m -» (A -» A; a -* amI~Construe T similarly. Then an M-homomorphism f:(A, M) -* (fi, M) is a natural transformation / : S -* T with component / : A -* B at the unique object 1 of the common domain category M. The diagram (1.4.2) of Definition 1.4.1 becomes /

for each element m of M. The natural transformation / : S -* T ¡s a natural isomorphism if and only if f:(A, M) -» (B, M) is an isomorphism of M-sets. D Example 1.4.3. For sets Yj and Y2, consider the functors FY :Set-> Set and Set- Set of Example 1.2.2, with respective object parts Z -> Z X Yx and Z Z xY2. Let g : Y, -» Y2 be a function. Then there is a natural transformation y:FY-* FYi with component yz : Z x Yj -» Z x Y2; (z, y) (z, yg). The diagram (1.4.2) of Definition 1.4.1, for a function / : Zx -2' becomes

rZl

>(*.yg)

(zf>y)<-^-* (~f.i/g) for z e Z , and y e Yj.

□

Example 1.4.4. Let ÍC be a commutative, unital ring. Let 5 : Mody -» Mod be the identity functor, and let T : Mod r -» ModK be the (covariant) com-

POSETS, MONOIDS, AND CATEGORIES

211

functors K\ ; K) : Mod„ -♦ Mod^ and Mod(cf. Exercise 1.2P). Then for each tf-module V, KK ;K): Mod£P the evaluation map of Exercise II 2.1E is the component at F of a natural transformation between S and 7. D posite

of the "functionar

Given two natural transformations a : S -» 7 and T : 7 -* U between functors S:C -* D, T:C -* D, and U:C -> D, a composite a ■ T or or : S -» U is defined by its components (O-T)X = O-XTX at objects X of C For a C-morphism f:X-* Y, the diagram

AS

AS

(°" T )A-

-»AT

AU

-» AU

/T

(1.4.3) YS

YS

►IT

(<^)i

■* y u

yu

in Z) commutes, so that or is also a natural transformation. This composition of natural transformations is associative. Given a functor S:C-*D, there is a natural isomorphism l s : S -» S with components lxs : AS -»AS at objects A of C Such a natural isomorphism l s or 1 : S -» S is called an identity transformation. If T : S -+ T is a natural isomorphism, then an inverse transformation T'1 :T -» S is defined, with component (r _1 )y = Ty1 at an object Y of T. Note that TT _1 = l s and T"V = l r . For small categories C and D, the set Cat(C, D) = (Z) c ) 0 of functors from C to D becomes the object set of a category Dc, a so-called functor category. For functors S : C -» D and 7 : C -» D, the class D C (S, 7) is the set of natural transformations between S and T. By (1.4.1), £>C(S, 7) is a subset of Set(Cn, Dx). Composition in Dc is the composition err of natural transformations. If C is small and D is large, one similarly obtains a large category Dc. Note that each functor S:C -* D is represented by a set, the graph {(/,/5) e Q x C^SI/e Cj} of its morphism part. Similarly, each natural transformation T : S -+ 7 is represented by a set, the graph {(A", TX) e C 0 X c c CQT\X e C0} of the function (1.4.1). Thus (D ) 0 is a class and (D )i is a class.

212

CHAPTER III CATEGORIES AND LATTICES

Example 1.4.5. Let M be a monoid. Then Example 1.4.2 shows that the large functor category Set** is (isomorphic to) the category M of Af-sets and M-homomorphisms. D Now suppose that there are functors Ä : ß -» C, S : C -» £>, T :C -* D, and U : D -» E, together with a natural transformation T : S -» 7\ Define TU:CQ-*EX\X~>(TXU:XSU-+XTU).

(1.4.4)

A diagram (1.4.2) for TU is obtained by applying the functor U to a diagram (1.4.2) for T. Thus TU becomes a natural transformation TU: SU -* TU, with components given by (1.4.4). Dually, define (1.4.5)

RT:BQ

-*D¿W*

(TWR

:WRS-+ WRT).

A diagram (1.4.2) for RT at a morphism / : W -» W of B is obtained as the diagram (1.4.2) for T at the morphism fR : WR -» WR of C. Thus RT becomes a natural transformation RT : RS -* RT, with components given by (1.4.5). EXERCISES 1.4A. Verify the claims in Example 1.4.4. 1.4B. Let K be a field, and let Mody " be the full subcategory of Mod^ consisting of finite-dimensional vector spaces over K. Let T < " be the restriction of the functor T of Example 1.4.4 to the subcategory Modr *". Show that the evaluation maps are the components of a natural isomorphism between the identity functor on Mod^ " and the functor TTÑ,;1)) *->(f:M-*N). Show that there are natural transformations 1 7 : 1 ^ -» FU with components r\A: A ^* A* and e:UF -* l Mon with "each component eu : M* -» M defined as the unique monoid homomorphism extending the identity

LIMITS AND LATTICES

213

(c) Verify the identities (VF)A(Fs)A

= 1AF

for each alphabet A and

(Ur,)„(sU)u-luu for each monoid M. 1.4E. Let (s,f):(A,M,R)^>(B,N,R') be an action morphism. Given a monoid homomorphism e : L -* M, show that there is an action morphism (es,ef):(A,L,eR)-*(B,N,R'). [Hint : Use Example 1.4.2 and (1.4.5) to define es.] 1.4F. Let D be a category with terminal object T. Let C be a small category. Define a functor TA : C -» D with object part C0 -* {T} and with morphism part C, -» {l r }. Show that TA is a terminal object of the functor category Dc. 1.4G. Let A' be a poset category. For a positive integer m, let m denote the poset category with Hasse diagram 1 < 2 < ••■ < m. Identify the objects of the functor category Xa- as the (m - l)-simplices in the poset (X,<,). 2. LIMITS AND LATTICES Semilattices were introduced in Section O 4.3 as idempotent, commutative semigroups. Given such a semigroup (5,- ), an order < is defined on S by (O 4.3.3). The partial order (S,<.) has the special property that each subset {a,b) of S has a greatest lower bound glb{a,¿>} given by the product a ■ b in the semigroup (£,-). Conversely, if (S, ¿ ) is a partial order in which each subset {a, b) has a greatest lower bound glb{a, b), then there is a semilattice (S,- ) whose product a • b is given by the greatest lower bound glb{a, b). A semilattice S may be realized as a poset category (5, <, ). The defining property of the greatest lower bound glb{a, b) or product a • b of two elements (objects) a, b of S may then be expressed diagrammatically in the category 5 : a* (2.1)

a■b

*b

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CHAPTER i n

CATEGORIES AND LATTICES

Given any object c with arrows c -* a and c -* b, there is a unique arrow c -» a-b such that (2.1) commutes. The top row of (2.1) says that a-b <.a and a-b <,b. The rest of the diagram represents the implication (c <. a and c <,b) =* c <,a-b. Now the diagram (2.1) for the product a-b in the category S is strikingly reminiscent of the diagram (O 3.2.4) for the direct product A x B of two objects A and B in the category Set of sets. One is thus led to the following Definition 2.1. Let Ax and A2 be objects of a category D. Then the product of the objects Ax and A2 is an object Ax XA2 of D together with morphisms TTX : Ax x A2 -» J4J and 7r2 : Ax x /1 2 -* A2 of £) known as projections, having the following universal property :

Í

for B e D0 and for each element ((/\ : fl -* Ax), (f2: B -> A2)) of £>(£, Ax) X £>(B, y42), there is a unique element (fx,f2) of D ( 5 , ^ ! X/t 2 ) such that ( / „ / 2 ) i r , = / , and ( / l f / 2 ) i r 2 - / 2 .

The universality property (2.2) is represented by the diagram ¿i

(2.3)

ITi

/I, X / l 2

■JT,

-»^4,

(/1./2)

/i

5 «-

S

-» 5

in D. D Example 2.2. (a) If AT is a poset category, then the product of two elements x,y of X is the greatest lower bound glb{x, y). Note that a pair of elements of a poset may or may not have a greatest lower bound (compare the antichains of Exercise O 4.3A). Thus in general, a pair of objects of a category may or may not have a product. (b) If Ax and A2 are sets, then the product Ax x A2 of Ax and A2 in Set is a direct product of Ax and A2. (c) Products in the concrete categories of Example 1.2.7 are given by the componentwise structure on the Cartesian product. For example, (O 4.8) describes products in the category Mon of monoids. D The dual Set"1* of the category Set of sets has a product of each pair Ax, A2 of objects, namely the disjoint union AXUA2 together with the insertions A¡ -*AX U A2. In general, the product in D op of two elements Ax, A2 of (D op ) 0 = D0 is called the "coproduct" Ax +A2 in D of the objects Ax, A2 of D. Thus the coproduct in Set is the disjoint union. Formally, one has the following:

LIMITS AND LATTICES

215

Definition 23. Let Ax and A2 be objects of a category D. Then the coproduct of the objects Ax and / 1 2 is an object Ax + A2 of D together with morphisms i, : Ax -+AX +A2 and i2 : A2 -* Ax + A2 of D known as insertions, having the following universal property : For 6 e D 0 and for each element ((fx : Ax -» 5 ) , (/ 2 : /1 2 -» B)) of £>(/!,, B) x D ( i 4 2 , f l ) , there is a unique element fx + / 2 of D ^ + ^ 2 , 5 ) s u c h t h a t ^ / ! + / 2 ) = / , and t 2 (/, + / 2 ) = / 2 . ! This universality property (2.4) is represented by the diagram Ax (2.5)

/. B

in D.

ij

* Ax +A2 *

t2

A2

/1+/2

-» B *-

/2

B

D

Example 2.4. (a) Let 5 be a unital ring, and let V, W be objects of Modc. Then the direct sum V © W, together with the insertions V -» V © W and W -* V © W, is the coproduct of F and W in Mode (b) Let (X, ^ ) be a poset category, with elements x and y. The coproduct of x and y in X is the /«wi upper bound lub{;t, y}, the element of X satisfying x <, lub{x, y}, y ^ lub{;t, y}, and ((* ^ z and y £ z) => lub{x, y) ^ z). (c) A coproduct in D°p of two elements ^4,, A2 of (£>op)0 = DQ is the product of Ax and A2 in D. D Let (5, + ) be a semilattice, written "additively," i.e. with + as the idempotent, commutative, and associative binary operation. Define a relation <, + or ^ on 5 by (2.6)

x <,y «=> x + y = y.

Then (5, ¿ ) is a partial order in which each subset {a, b) has a least upper bound lub{a, b} = a + b. Conversely, given such a partial order (5, ^ ), one may set a + b : = lub{a, ¿} to obtain a semilattice (5, + ). A semilattice (5,+) ordered according to (2.6) is called a join semilattice (S, + , <, + ), while a semilattice (5, • ) ordered according to (O 4.3.3) is called a meet semilattice

(S,,<0-

A lattice may be defined order-theoretically as a partial order (L, ^ ) in which each subset {a, b) has both a greatest lower bound glb{a, b) = a-b and

216

CHAPTER III

CATEGORIES AND LATTICES

a least upper bound lub{a, b) = a + b. Thus both (L, • ) and (L,+) are semilattices. If L is ordered as a meet semilattice from (L, • ), and as a join semilattice from (L,+ ), then the two orders agree, yielding the original order (L, ^ ) . Thus a lattice may be characterized as a set (L, + , • ) with two semilattice operations + and -, such that the join semilattice order ^ + and meet semilattice order ^ agree. It is sometimes convenient to denote the product x • v by the juxtaposition xy. Moreover, in compound expressions, the product or meet x • v or xy will bind more strongly than the coproduct or join x + y. Aside from this notational convention, the concept of a lattice is self-dual. Note that duality on a lattice interchanges the roles of meet and join. Completing the transformation from order-theoretic to algebraic characterizations, one obtains the following: Proposition 2.5. Let (L, +, • ) be a set with two semilattice operations related by the absorption laws (2.7)

x-y+y=y

and (2.8)

x-(x+y)

= x.

Then the meet semilattice and join semilattice ordering on L agree, yielding a lattice ( L , ^ ) . Conversely, if (L,^) is a lattice, then the absorption laws are satisfied. Proof. If (2.7) is satisfied, one has x^y=>xy=x=»x+y=x-y+y = y =» x ^ +y. Dually, (2.8) yields x <, +v =» x <. y. Thus the meet and join semilattice orderings agree if both absorption laws are satisfied. Conversely, in a lattice (L, <.), one has y = lub{glb{;c, y), y), so (2.7) holds. Dually, (2.8) holds. D Given the algebraic characterization Proposition 2.5 of lattices, one may apply the usual algebraic constructions to lattices. For example, the direct product of two lattices (L, + , ) and (Af, + ,-) is the set L x M with componentwise join and meet. A sublattice of a lattice (L, + , • ) is a subset S of L such that (5,+) is a subsemigroup of (L,+) and (5, • ) is a subsemigroup of (L, •). A congruence a on a lattice (L, + , ) is an equivalence relation a on L that is a sublattice of (L2, +, • ). A lattice homomorphism f : (L, +, • ) -» (M, +, • ) between lattices (L, +, • ) and (M, +, • ) is a function f:L-*M that gives semilattice homomorphisms f:(L, + ) -* (M, + ) and / : (L, • ) -» (M, • ). The concrete category LQ of lattices has the class of all lattices as its object class Latn. Given lattices L and M, the class Lat(L, M) is the set of lattice homomorphisms from L to M.

LIMITS AND LATTICES

217

EXERCISES 2A. Let D be a category in which any two objects have a product. Let Av A2, A3 be objects of D. Exhibit isomorphisms T : Ax XA2 -> A2 XAX and (Ax X A2) X A3 = Ax X (A2 X A3). 2B. Let A and ß be sets. Show that (A U 5)* is the coproduct in the category Mon of the free monoids A* and B*. 2C. (a) Let T be a terminal object of the category D. For each object A of D: (i) show that the product T X A exists; and (ii) show that there is an isomorphism A = T xA. (b) Formulate and do the dual of (a). 2D. Does the category Pfn (cf. Example 1.1.5) have a product for each pair of objects? 2E. For each of the following partial orders (L, <, ), determine whether L is a lattice : (a) the intersection of the linear orders 1 < 2 < 3 < 4 < 5 < 6 and 1<3<2<5<4<6; (b) the intersection of the linear orders 1 < 2 < 4 < 3 < 5 < 6 and 1 < 3 < 2 < 5 < 4 < 6 [cf. Proposition O 3.5.4(b) and Exercise O 3.5E]. 2F. (a) For a set B, show that the set &< a(B) of finite subsets of B forms a lattice (&>< U(B), ç ) under the relation of containment, (b) Define a subset X of the set B to be cofinite if its complement B - X is finite. Let 3s < "(B) denote the set of cofinite subsets of B. Show that (a* < "(B), ç ) is a lattice. 2G. Let Q be a quasigroup, let Sb Q be the set of subquasigroups of Q, and let Sb > 0 Q be the set of non-empty subquasigroups of Q. Let ç be the containment relation. (a) Show that (Sb Q, ç ) is a lattice. (b) Show that (Sb >0 Q, ç ) is a join semilattice. (c) Give an example of a quasigroup Q for which (Sb > 0 ß, ç ) is not a lattice. 2H. Let (L, + , • ) be a set with two semilattice operations. Let (L,+) be the join semilattice with Hasse diagram 1 -» c -» 0, and let (L, • ) be the meet semilattice with Hasse diagram c -> 1 «- 0. Show that (L, + , • ) satisfies the absorption law (2.7), but not the absorption law (2.8). [See J. Gahiszka, "On Bisemilattices with Generalized Absorption Laws, I," Dem. Math. 20 (1987), 37-43.] 21. State and prove the First Isomorphism Theorem for lattices. 2J. Show that any two objects of Ljt have a product.

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CHAPTER III CATEGORIES AND LATTICES

2K. Consider the poset C with Hasse diagram (1,0)

»(2,0)

(0,0)

»(1,1)

K2.1)

Let D be the poset 0 < 1 < 2. Show that C and D are lattices, and that F :C -* D; (x,y)-*x is an order-preserving map that is not a lattice homomorphism. 2L. Let F : C -» D be a bijection between lattices C and D such that F : (C, ¿ ) -» (/>,£) is an isomorphism in Pos. Show that F is a lattice homomorphism. 2M.(a) Show that the forgetful functor U : Mon -» Set with object part (M, • ,1) -» Af preserves products, in the sense that ((M, • ,1) X (N, ■ ,l))U = MxN. (b) Show that U does not preserve coproducts. 2N. Let C be a small category, and let D be a category in which each pair of objects has a product. Show that any two objects Sv S2 of the functor category Dc have a product 5, X S2 defined pointwise by if:X — YXS} X S2) = d r , / s ' , i r 2 / s 0 : XS, X XS2 -* YS1 X YS2 for each C-morphism f:X-*Y. [Hint : The projection ir' : 5i x 5 2 -» S¡ in Dc has component 7r¿ = 7r, : A"5j x ¿ S 2 -* XS¡ at an object X of C]

2.1. Products and Coproducts For a set B, the set âB of elements has a greatest lower bound, namely C\jmJB¡. If {A¡\j e / ) is a multiset of sets, then one may construct the product Y\.¡SJA¡, along with projections TT¡ : riy e y/l 7 -* A¡ for each i in /, such that, given a multiset (f¡ :B -> Aj\j e / > of maps, there is a unique map f-B-*nj6jAj with /ir¡ = /¡ for each i in / (cf. Section O 3.2, including Exercise 3.2Q). For a two-element set /, the greatest lower bound C\jeJBj and product njeJAj are instances of the two-factor categorical product of Definition 2.1. One is thus led to the following, more general definition: Definition 2.1.1. Let / be a set, and let D be a category. Let A : J -* D0; j -» Aj be a function. Then the product ir or irA of A is an object njmJAj of D together with morphisms TT¡ : Y\je]Aj -» A¡ (known as projections) for

LIMITS AND LATTICES

219

each i in /, having the following universal property:

{

For each object B of D and element < of YljeJ D(B, A¡), there is a unique element n.jBj
jA¡)

such that
The universality property (2.1.1) is represented by the diagram

UAi ¿ILA * •

in D.


B

B

D

By the usual abuse of notation, the object TljejAj itself is often called the product, leaving tacit the role of the projections. Example 2.1.2. (a) Let D be one of the concrete categories of Example 1.2.7. Then for any function A : J -» D0 (with domain a set / ) , the product object YlieJAj is given by the product set njeJAjU with componentwise structure. "Componentwise structure" means precisely that the projections v¡ are morphisms of D. (b) Let {X, i ) be a poset, construed as a category. Let A : / -» X; j <-* Xj be a function. Then the product irA is the greatest lower bound n{jcy|; e /} of the set {x¡\j e / } , an element Yl{xj\j e J) of X with the two properties: (0 V i e / . n m / e / } £ * , . ; (ii) (y e X and Vi e /, y <, x¡) =» y ¿, TlU^j G /}. Note how (i) corresponds to the existence of the projections TT¡, while (ii) corresponds to the existence of the horizontal arrows in (2.1.2). As already seen for the case |/| = 2, such a greatest lower bound may or may not exist. (c) Let D be a category with a terminal object T. Then T is the product of the function 0 : 0 -» D0. All but the top line of the diagram (2.1.2) disappears, while the existence and uniqueness of the arrow in the top line follows from the Definition 1.3.3(a) of a terminal object. (d) For an infinite set B, consider the poset (^a< a(B), ç). As observed at the beginning of the section, a function A: J -+âB< m(B)

220

CHAPTER III CATEGORIES AND LATTICES

has no upper bound, i.e. the poset category (&>< a(B) has no product. D Products in the dual D op of a category D are called "coproducts." Formally, one has the following definition, generalizing Definition 2.3. Definition 2.1.3. Let / be a set, and let D be a category. Let A:J-*D0; j -> Aj be a function. Then the coproduct t or iA of A is an object T.jeJAj or lljejAj of D together with morphisms i,: Ai¡-» T.ieJAj (known as insertions) for each i in /, having the following universal property:

Í

For each object B of D and element < of njeJD(Aj, there is a unique element Eje/
B),

D(T,jeJAj,B)

such that (p¡ = 4,(E;ey
The universality property (2.1.3) is represented by the diagram

(2.1.4)

in D.

Î

D

Again, one sometimes refers to T,ieJAj alone as the coproduct, not explicitly mentioning the insertions. The notation U.jeJAj for the coproduct is preferable in categories such as Mod; where one wishes to reserve the notation E ; 6 y ^4 ; for other purposes, such as the submodule of a module X generated by the union UM;-l/ ^ J) of submodules A¡ of X. Example 2.1.4. (a) In the category Set, the coproduct Ey g/ y4 ; of an arbitrary function A : J -» Set0 from a (smaÏÏ) set / is given by the disjoint union construction of Exercise O 3.2P. (b) In the category Modz of abelian groups, the coproduct UjeJAj is given by the construction of II 1.1 as a subgroup E ie7 ^4.i, of nieJA¡ [cf. (II 1.1.5)]. (c) For a function A : / -> Setn; j -» A¡ from a (small) set /, the coproduct in Mon of the function A : / -» Monn; j -* A* is the free monoid (X.jejAj)* on the disjoint union T.¡SJA¡ of the alphabets Aj.

LIMITS AND LATTICES

221

(d) Let (X, ^ ) be a poset, construed as a category. For a function A : J -» X; j -* Xj, the coproduct iA is the least upper bound Z(xj\j e /} of the set {Xj\j e /}, defined dually to the greatest lower bound. (e) Let D be a category with an initial object _L . Then _L is the coproduct of the function 0 : 0 -» D 0 . D For a poset category (A!", ^ ) , the existence of the product (greatest lower bound) of any subset is equivalent to the existence of the coproduct (least upper bound) of any subset. Proposition 2.1.5. Let (X,<,) be a poset, construed as a category. Then the following conditions are equivalent : (a) For each function A : J -» X; j -» x¡, the greatest lower bound n{jCy|y e /} exists; (b) For each function A:J-*X;j-* x¡, the least upper bound T,{x¡\j e /} exists. Proof. Suppose that (a) holds, and that A:J-+X;j-*Xj is a function. Define U = {u e X\Vj e /, x¡ <, u], i.e. the set of upper bounds of JA. Consider the inclusion function I:U -*X; u -» u. By (a), the product TIU exists. Consider x¡ e JA. By Example 2.1.2(bXii), (Vu e U, x¡ <. u) =» x¡ <. TIU, so that YIU is an upper bound for JA. Suppose that u is an upper bound for JA. Then by Example 2.1.2(bXi), one has TIU ^ u. In other words, TIU is the least upper bound of JA. Thus (b) holds. The proof (b) =» (a) is dual. O A poset satisfying the equivalent conditions of Proposition 2.1.5 is called a complete lattice. The binary product • or meet in a complete lattice (X, <) yields a monoid (X, • ,1) whose identity element is the empty product or terminal object [cf. Exercise 2C(aXii)]. Dually, the binary coproduct + or join in the complete lattice (X, ^ ) yields a monoid (X, + ,0) whose zero element is the empty coproduct or initial object. A lattice (X, +, ■ ) is bounded above if the semigroup (X, • ) is a monoid (X, ■ ,1), and bounded below if the semigroup (X,+) is a monoid (X, + ,Q). A lattice (X, + , ■ ) is bounded if it is bounded both above and below. Thus a bounded lattice (X, + ,-,0,1) comprises monoids (X, + ,0) and (X, ■ ,1). A bounded lattice homomorphism f:(X, + , • ,0,1) -» (Y, + , • ,0,1) is a lattice homomorphism with 0/ = 0 and 1/ = 1, i.e. a function / : X -* Y that gives monoid homomorphisms / : (X, + ,0) -► (Y, + ,0) and / : (X, ■, 1) -* (y, •, 1). Note that a finite lattice is complete (Exercise 2. ID). Example 2.1.6. For a set B, consider the power set (2B, ç ) ordered by containment. The terminal object of 2B is B. If A: J -» 2 B ; ; ->/iy is a function, then the product irA or njeJAj is the intersection n y e y / l 7 . Thus (2B, ç ) is a complete lattice. D

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Example 2.1.7. Let Q be a quasigroup, and let (Cgß, ç ) be the set of congruences on ß ordered by containment. The terminal object of Cgß is Q2. If A : / -» Cgß; j -* otj is a function, then the product n ; S / a y is the intersection C\jej a¡- Thus (Cgß, c ) is a complete lattice. D One of the most striking properties of complete lattices is given by the Tarski Fixed-Point Theorem 2.1.8. Let (X, <,) be a complete lattice, and let T be a covariant functor (i.e. monotone function) T : X -* X. Then the induced poset {(Fixi-Y/T», ^} is a complete lattice. In particular, the action (X,(T)) does havefixedpoints. Proof. Let A be a subset of Fix(X,(T)). Set B = {b e X | Va e A, a <. bT ^ b). Consider the greatest lower bound g of B in X. Now for b in B, one has g ^ b, whence gT £ bT £ b. Thus gT :£ g. Next, consider the least upper bound / of A in X. Then (V¿> e B, a <. bT) =» (Vft &B,l <,bT <.b) => / ^ g. Thus g is an upper bound for A in X, i.e. Va e ,4, a <, g. Applying T yields Va e y4, a 7 = a <,gT £ g, whence g s 5 , Applying T again yields Ma^A,aT = a <, gT2 <, gT, whence gT e J5. Thus g ^ gT, and so g!T = g e FixiAT/r». In other words, g is an upper bound for A in (Fix(X,(T)), <,). If u is any upper bound for A in (¥\x(X,(T)),<,), then (Va e A, a <, uT = u)=»ueB=*g<,u. Thus g, the greatest lower bound for B in X, is the least upper bound for A in Fix(A',
-* ( 2 " , ç ) ; Y ~ > A - Y.

Their composite T:(2A,c) -* (2A,Q) is a functor on the complete lattice (2A, Q)—cf. Example 2.1.6—and thus has a fixed point Alt i.e. Ax = A{T = ^,SiS 2 5 3 5 4 . Now set A2 =A1S1S2S3, B2 = A^^, and 5 , =^4,5,. (b) If / : A -» B and g : B -» A inject, then (f:Ai^> BJ U(g _ 1 : A2 -> B2) is a bijection from A to B. D

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EXERCISES 2.1A. Verify the claims of Example 2.1.2(a) for the concrete categories Mon andQ. 2.1B. (a) Give a precise definition of the least upper bound T.{x¡\j e /} of a subset {Xj\j 6 /} of a poset (X, ^ ) . (b) What is the least upper bound of the empty subset of a poset (X, <;)? When does it exist? 2.1C. Verify the claims of Example 2.1.4. 2.1D. Show that a finite lattice (X, +, • ) is complete. 2.1E. Show that the set Eq B of equivalence relations on a set B, ordered by containment, forms a complete lattice (cf. Exercise O 3.3B). 2.1F. Let X and Y be countable, infinite sets [cf. Exercise 1.1 A(b)]. Prove X=Y. 2.1G. Let X = {0} U {n-110 < n e Z}. Let Y = {-1} U X. Consider X and Y with the orders induced from (R, ^ ). (a) Show that X and Y are complete lattices. (b) Define F: X -» Y by xF = if x = 0 then - 1 else x. Show that F is a bounded lattice homomorphism. (c) Exhibit a subset {x¡\j e / } such that U{xjF\j e / } # [I~IU I; e /}]F. 2.1H. Let (X,;£) be a complete lattice. Define r : ( 2 * , ç ) -* (2*,ç); y - » {Ey}D (cf. Example 1.2.3). Show that T is a functor, and show that the lattices (Fix(X,(T)), + , • ) and (X, + , ■ ) are isomorphic. 2.2. Slice Categories Let C be a category, and let x be a point of C. One often wishes to consider the category C "relative to" the object x, or to "take a slice of C from the standpoint x." To this end, one constructs a new category C/x, called the slice category of C-objects over x, as follows. An object of C/x is a C-morphism p : a -* x, i.e. an arrow of C with head x, an element of d^{x). A morphism / : (p : a -» x) -* (q : b -* x) of C/x is a C-morphism / : a -» ¿> such that the diagram / a —i—* 6 (2.2.1)

p

ç

x === * in C commutes. If g : (q : b -» x) -» (r : c -» x) is a second (C/x)-morphism, then the equation p = fq = f(gr) = (Jfc)r in C shows that the composite

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C-morphism fg:a-*c yields the composite (C/jc)-morphism fg:(p:a -»x) -* (r:c -* x). The identity at p:a-*x in (C/x) is the (C/r)-morphism l a : (p : a -» x) -* (p : a -* x). Example 2.2.1 (Down-sets). Let (X, ^ ) be a poset category. For an element x of X, consider the down-set xD = {y e X\y ^ x) (cf. Example 1.2.3). Then the slice category X/x is isomorphic to the down-set (xD, ^ ) with the partial order induced from (X, ^ ) . The isomorphism is given by the forgetful functor X/x -» xD with object part (y -» x) -* y. D Example 2.2.2 (Augmentations of Rings). An augmentation of a unital ring R is a unital ring homomorphism e\R-*I. from R to the ring of integers. For example, the map e:Z[T]-*Z;£PiT'~

£

i-O

Í-0

Pi

is an augmentation of the integral polynomial ring Z[T]. Then augmentations of rings are precisely the objects of the slice category Ring/Z of unital rings over Z. D Example 2.23 (Subobjects). Let C be a category. The composite of two monomorphisms in C is again a monomorphism in C (cf. Exercises 1.3D and 1.3G). Moreover, identity morphisms in C are monomorphisms in C. One obtains a subcategory Mon(C) of C with Mon(C)0 = C0 and with Mon(C)i the class of monomorphisms in C. Now let x be an object of C. Consider the slice category Mon(C)/x. Objects of Mon(C)/x are called subobjects of x, and an isomorphism of subobjects of x is an isomorphism in the category Mon(C)/x. D For an object x of a category C, the dual slice category (C o p /r) o p is called the slice category x/C of C-objects under x. Thus an object of x/C is a C-morphism l:x-*a, i.e. an arrow of C with tail x, an element of dô1{x}. A morphism / :(l : x -» a) -> (m : x -» b) of * / C is a C-morphism f:a-*b such that the diagram x ===== x (2.2.2)

m

/ • t

*

a —T—» 6

in C commutes, while the composite of / with the (*/C)-morphism g : (m : x -» b) -» (« : * -» c) is _/£ : (/ : x -» a) -* (n : x -* c) given by fg : a -* c in C.

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Example 2.2.4 (Up-sets). If x is an object of a poset category (X, ^ ) , then the slice category x/X is the up-set {y e X\x £ y) of x with the partial order induced from (X, ^ ) . (Warning: Elements of the up-set x/X are A"-objects under x. Elements of the down-set X/x are A'-objects over x. The vertical orientation implicit in the terminology "over *"and "under x"is the reverse of that used in Hasse diagrams and implicit in the terms "down-set" and "up-set".) D Example 2.2.5 (Commutative Algebras). Let CRing be the concrete category of commutative, unital rings and homomorphisms. For an object K of CRing, (codomains of) objects of the slice category K/ CRing are commutative .K-algebras (cf. Proposition II 3.1). □ Example 2.2.6 (Partial Functions). Fix a singleton {*}. Then the slice category {*}/ Set embeds into the category Pfn of partial functions (cf. Examples 1.1.5 and 1.2.9) via a functor F : {*T7~Set -> Pfn whose object part is (/:{*}-y)^y-{*}/. D Section 1.3 introduced one of the prime applications of category theory: the specification of mathematical constructions as initial (or dually, terminal) objects in certain categories. By Proposition 1.3.4 and Corollary 1.3.5, such specifications are unique, to within isomorphism. Products and coproducts are typical mathematical constructions. By taking subcategories of slices of certain functor categories, one obtains new categories having coproducts as initial objects, and products as terminal objects. Consider a category D, a set /, and a function A : J -> D 0 . The coproduct iA will be expressed as an initial object. The first task is to make thefunction A:J-*D0 into a functor. The set J determines an antichain ( / , / ) , which may then be construed as a discrete (poset) category / (cf. Exercise 1.2B). The function A:J-*D0 becomes the object part of a uniquely specified functor A: J -» D. Thus A is a point of the functor category D1. Note that each point of D1 is specified completely by its object part / -» D 0 , since the category / has no non-identity morphisms. The functor category D1 contains a subcategory DA that (for / + 0 ) is an isomorphic copy of the category D, the isomorphism being given by a diagonal functor A : D -* DA. The object part of A sends an object B to the constant functor BA:J-*D with object part SA : / -» D 0 ; j -» 5. The morphism part of A sends a D-morphism f:B'-*Bto the natural transformation /A : B'A -» BA whose component at each element j of / (i.e. at each object j of the category / ) is the same D-morphism f:B'-*B. Note that verification of (1.4.2) for /A is trivial. Now consider the slice category A/D1 of Dy-objects under A. An object of A/D1 is a natural transformation from A to a functor J -* D. Let A/DA be the subcategory of A/D1 whose objects are natural transformations

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object i of / is a D-morphism A)-morphisms /A : dp' : A -* B'A) -*(
fp'i

A

*B

in D commutes. One may readily verify that A/DA is indeed a subcategory of A/D} (Exercise 2.21). It is called the category of constant functors under A. The universality property (2.1.3) for the coproduct iA of A is then expressed precisely by the following: Proposition 2.2.7. Let D be a category, and let J be a set or discrete category. Let A:J-*Dbea functor. Then the coproduct tA of A is characterized as an initial object of the category A/DA of constant functors under A. In particular, the coproduct object T,jeJAj of D is unique to within isomorphism. Proof. If the coproduct iA exists, then (2.1.4) exhibits a unique {A/DA)morphism E - e / ^ A from iA : A -* C£,jeJAj)A to each (y4/DA)-object (p : A -* BA. Conversely, given an initial object B"A), by Corollary 1.3.5. Then f'.B'-* B" is a D-isomorphism between the two coproduct objects B' and B". D Dually, one may construct the category of constant functors over A, a subcategory DA/A of DJ/A whose objects are natural transformations of the form /A : B'A -» BA for D-morphisms / : B' -» B with
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227

2.2B. Let T be a terminal object of a category C. (a) Show that C is isomorphic to the slice category C/T. (b) Show that C is isomorphic to the slice category (C/T)/(T -* T), (c) What is the terminal object of the slice category (C/T)/(T -» T)l 2.2C. (a) Verity that 1[T] -* 1; L^opF* -» £?_„/>,• i s a u n i t a l r i n g h o m ° morphism. (b) Show that the polynomial ring Z[T] may be augmented by the map e:HT]->Z;LUP,T'*+Po2.2D. Let Z[S3] denote the set Set(S3,Z) of functions from the symmetric group S3 to the ring Z, with pointwise subtraction. Define a binary operation * of convolution on Z[S3] by f*g(T)-

E

UV-T

f(U)g(V)

for each T in S3. Show that (Z[S 3 ],-, *) is a unital ring, and that s:f-* T.TeS f(T) is an augmentation of Z[S3]. 2.2E. Let x be an object of a category C. Define a relation <x on (Mon(C)/*)0 by i'o-z *» 3 ( / : >> -♦ z) e (Mon(C)/jc),. (a) Show that a is a pre-order. (b) Consider the equivalence relation a on (Mon(C)/x)0 given by yaz *» yaz and zo-y (cf. Exercise I 1.3H). Show that a is the relation of isomorphism of subobjects defined in Example 2.2.3. (c) Consider the order relation ^ on (Mon(C)/x)Ô given by y" ^ z" «=» yaz (cf. Exercise I 1.3H). For a set £, show that there is an isomorphism (,3»{B), c) -» ((Mon(Set)/fi)J, ¿ ) ; X -* (X -* B)a. 2.2F. Let (L, + , • ) be a lattice, construed as a poset category (L, ^ ) . For elements x, y of L, prove (a) (L/*) 0 O (L/y) 0 = {L/xy\; (b) (x/L) 0 n (y/L)Q = ((* + y)/L) 0 . 2.2G. Complete the specification of the functor F of Example 2.2.6, and determine its inverse. 2.2H. Let A : D -* D0 be the diagonal functor from a category D to the functor category D0. Determine the category D0 and its subcategory DA. 2.21. Verify that (2.2.3) leads to a subcategory A/DL of A/D1. 2.2J. Let A and / be sets. Show that the product of the object A A of Set7 is the evaluation natural transformation ev:Set(/,>l)A ->.4A whose component at the element j of / is ev,.:Set(/,/l)

-*A;f~jf.

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2.2K. Let x be an object of a category C. (a) Show that l x : x -* x is a terminal object of the slice category C/x. (b) Show that lx : JC -» JC is an initial object of the slice category x/C.

2.3. Equalizers and Pullbacks The Definition 2.1 of the product of two objects in a category was obtained as a generalization of the Definition O 3.2.2 of the direct product of two sets. This section examines the construction (O 3.3.7) of the natural projection of a set A onto its quotient A" by an equivalence relation a, formulating the construction categorically so that it may be applied readily in other contexts. The equivalence relation a on the set A is a certain subset of the direct product AX A. The direct product A X A or A1 comes equipped with the two projections TT¡ : A2 -» A\(av a2) •-» a¡ for í = 1,2, each of which restricts to the subset a of A2. In the diagram IT2

nat a

7T,

(2.3.1)

■* A

nat a

* A",

one has 77-^nat a ) = 7r2(nat a). In other words, the diagram (2.3.1) commutes. Moreover, given a function / : A -* B with irxf = ir2f, there is a unique, well-defined function / : A" -* B; x" -» xf such that (nat a)f = f. One is thus led to the following: Definition 2.3.1. Let D be a category. /i

(a) A parallel pair Ax ={ A2 of arrows in D is a pair (/ 1( / 2 ) of arrows of D Si

with common tail Ax and common head A2. /i

(b) Let Ax = M 2 be a parallel pair of arrows in D. Then the coequalizer h of the pair (/ 1( f2) is (an object Q of D together with) a morphism q : A2 -» (3 of £>, satisfying f\q=f2q and the following universal property. (2.3.3)

I For each D-morphism f:A2-*B with /, / = / , / , there is 2 . m _m> 1 a unique element f of D(Q, B) such that ç / = / .

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229

The universal property (2.3.2) is represented by the commuting diagram

(2.3.3)

Q

—r-»S

B in D.

D

As in the case of products, one often refers to the object Q alone as the coequalizer, suppressing the (important) role of the morphism q. g

q

Example 2.3.2 (Cokernels). Let Ax -*A2-*A3 -» Obean exact sequence of abelian groups. Consider the pair (g,0) of homomorphisms from A¡ to A2. Note that gq = 0 = Oq. Suppose there is a homomorphism / : A2 -* B with gf = 0 / = 0. Then Atg ^ Ker/, so there is a well-defined composite homomorphism / : Ai = A2/Axg -+A2/Kei f s A2f "-> B with qf = f. In other words, q : A2 -* A3 is the coequalizer of the pair (g, 0) in the category Mod,. /i

Conversely, let Ax ^XA2 be a parallel pair of arrows in Mod?. Then their fi coequalizer is the projection A2 -» CokeK/i - f2) (Exercise 2.3B). D Example 233 (Presentations of Groups). Let X — {xu x2) and Y = {yl,y2,yi}- Define a group homomorphism ff:YG-*XG between free groups, determined via (I 1.4.8) by the set map / , : Y -» XG; y, <-* x2, y2 -» x\, y3 -» xlx2x1. Similarly, define f2 : YG -» XG from f2:Y-> XG; y\ "-* If yi *"* 1» y* *~* x2xxx2. Then the coequalizer in G£ of the pair (f°, f2) of group homomorphisms is the group homomorphism q : XG -» S3 onto the symmetric group given by xx -» (12), x2 <-* (23). Indeed, fxq = f2q. Suppose / : XG -* Q is a a group homomorphism with /,/' = f2f. Set Í = * , / e ß and t = x2f <= Q. Then s2 =yJJ = yj2f = 1, t2 = y2fj = y2f2f = 1, and (sí) 3 = ÍÍJÍSÍ = (sfsXtef)-1 = iyifJXy3f2fYi = 1, whence there is a group homomorphism / : S3 -» ß ; (12) -» s, (23) -» f (Exercise I 3.21) satisfying 9 / = /. In general, set X = {JC„ . . . , x„) and Y = {y„..., yr}. Then the coequalizer of a pair YG 3 XG of group homomorphisms is called the group with presentation (xv ...,x„\yj1= yj2, ...,yrf1= yrf2) ^ The relation {(y(/„ y¡f2)\\ <,i ^ r) on XG is called the set of relationslmposed by

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the presentation, while the set {x 1 ( ..., x„) is called the set of generators in the presentation. D Examples 2.3.2 and 2.3.3 illustrate the specification of algebras as coequalizers: modules as cokernels in Example 2.3.2 and groups by presentations in Example 2.3.3. The precision of such specifications depends on the uniqueness of coequalizers (to within isomorphism). This uniqueness is demonstrated by exhibiting coequalizers as initial objects in suitably contrived categories, as was done for coproducts in Proposition 2.2.7. Thus cokernels and group presentations become typical examples of the theme broached in Section 1.3: the specification of mathematical constructs as initial objects. f\ The first task is to describe parallel pairs Ax zXA2 of arrows in a category h D in terms of functors. As a directed graph T, a parallel pair of arrows has the form •=£ •. Technically, one considers the directed graph T = (r 0 , r „ d0, dx) = ({1, 2}, {1, 2}, {1, 2} - {1}, {1, 2} -* {2}), i.e. the pair (AX, 2 -* A2 and with morphism part determined by {1,2} -» Dx; 1 -* fx, 2 -* f2. In other words, parallel pairs of arrows in D become objects of the functor category DJ. Given an object B of D, one has a constant functor BA :J-*D with object part 1 -» B, 2 -» B and with morphism part determined by {1,2} -* Dx, 1 —» 1 B , 2 —» 1 B . Given a D-morphism f:B'-*B, there is a natural transformation /A : BA -» BA whose component at each of the two objects of / is the same D-morphism f:B'-*B. Note that (1.4.2) for /A reduces to \B.f = f\B. One obtains a subcategory DA of the functor category DJ as the image of the diagonal functor A:D-*D' with object part £>0 -» (DJ)Q; B -* BA and morphism part Dx -> (D'\; (f:B'-*B)~ (/A : B'A -» BA). Now consider the slice category A/D1 of Dy-objects under the functor A:J -» D corresponding to a parallel pair AX^XA2 i

h

in D. Consider the

subcategory A/DA of A/D1 whose objects are natural transformations

2 : A2 -» B. The diagram (1.4.2) in D corresponding to the morphism 1:1 -* 2 in / yields <¡PJ1B = fxsp2. The diagram (1.4.2) corresponding to 2:1 -» 2 in J yields (px\B =f22 = y are just those natural transformations /A : 5'A -> BA for / : 5' -» B in D, such that, for each object i of /, the

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231

diagram A, —

A,

(2.3.4) B'

T-*

f

B

commutes. In fact, if (2.3.4) commutes for i = 2, i.e. s o that (2.3.4) automatically commutes for i = 1. The diagram (2.3.3) then shows that the object tA : A -* QA of A/DA determined by the coequalizer q : A2 -* Q is initial in A/DA. Conversely, an initial object i : A -* QA of A/DA yields a coequalizer i2 '■ A2 -» Q of the parallel pair (1A,2A) of morphisms in D. Summarizing, one obtains for coequalizers the following analogue of Proposition 2.2.7 for coproducts. Proposition 2.3.4. Let D be a category, and let J be the path category of the directed graph • 3 ■ or ({1,2} -» {1}, {1,2} -» {2}). Let A:J-» D be a functor. Then the coequalizer i2 : A2 -* Q of the parallel pair (1A, 2Ä) of arrows in D is characterized as an initial object i : A -* QA of the category A /DA of constant functors under A. In particular, the coequalizer object Q of Dis unique to within isomorphism. O Dual to coequalizers, one has the concept of the equalizer E or e : E -» Ai f\ of a parallel pair Ax =XA2 of arrows in a category D. Thus efl = ef2, and if h ff\ = ffi f°T a ö-morphism f:B -» Ax, then there is a £)-morphism f : B -* E with fe = /. Using the same category / as for equalizers, one may consider the slice category DJ/A and its subcategory DA/A of natural transformations to A from constant functors, the category of constant functors over A. One then obtains the following dual of Proposition 2.3.4. Proposition 2.3.5. Let D be a category, and let J be the path category of the directed graph ({1,2} -♦ {1}, {1,2} -> {2}). Let A: J -» D be a functor. Then the equalizer irf : E -* Ax of the parallel pair (ÍA, 2 A) of arrows in D is characterized as a terminal object IT : EA -* A of the category DA/A of constant functors over A. In particular, the equalizer object E of D is unique to within isomorphism. D /i

If AxzXA2 is a parallel pair of arrows in Set, then their equalizer h — e : E -» Ay certainly exists. One may take e to be the inclusion in Ax of the

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subset E = {a e A x \af¡ = af2). Then if ffx = ff2 for a m a p f : B - * A x , one has Bf £ E, so the restriction / : B -* E of the m a p / : B -+ A x satisfies fe = f. In fact, this construction works in any of the concrete categories of Example 1.2.7. T h e particular case of M o d y is worth considering in detail, as it offers a dual to Example 2.3.2 (Exercise 2.3K). Slice categories were used in Propositions 2.3.4 and 2.3.5 to demonstrate the uniqueness of coequalizers and equalizers. Conversely, one may use equalizers and coequalizers to determine properties of slice categories, namely the existence of products in slices C/x of a category C with products, and dually the existence of coproducts in slices x/C of a category C with coproducts. Let C be a category, and let x be an object of C. Let p x : a, -» x and p 2 : a2 -* x be objects of C/x. Let iri :alXa2-> a¡ be the product of ax and a2 in C. Let e : a, xxa2 -* a, X a 2 be the equalizer of the parallel pair (TTIPX, ir2p2) of C-morphisms. Set p = ev1p1 = eir2p2 : ax Xxa2 -» x. T h e n one may readily verify the following (cf. Exercise 2.3L). Proposition 2.3.6. The object p : ax Xxa2 -* x, with projections etr¡ :(p:al Xx a2 -» x) -» (/>, : a, - » * ) , « the product of the two objects p¡ : a¡ -* x (i = 1,2) in the slice category C/x. D In t h e category C, the object ax xxa2, together with the maps evi : ax xxa2 -> a¡, is called the pullback of the pair (pu p2) of C-morphisms with common codomain x. Such a pullback is described by the commuting diagram a, X, a2 (2.3.5)

eir2

► a2

Pi

etr.

Pi

-» x

in C. Taking / to be the poset category with Hasse diagram 1 -* 0 «- 2, one may characterize the pullback of ax -» x *- a2 as a terminal object in the category CA/A of constant functors over A , a subcategory of the slice category CJ/A, for the functor A : J -* C with morphism part ((1 -» 0) -* {px : ax -* x)), ((2 -» 0) -» (p2 : a2 -* x)) (Exercise 2.3M). Duals of pullbacks are called pushouts. They provide coproducts in slice categories x\C. This section began by considering equivalence relations. It concludes the same way.

LIMITS AND LATTICES

233

Example 23.7. Let / : A -* B be a set map. Then its kernel is the pullback ►A

ker /

f .

/

D

- B

EXERCISES 2.3A. Let q : A 2 -» Q be the coequalizer in Set of a parallel pair

h AX^A2

of functions. Show that Q is isomorphic to the quotient of A2 by the transitive closure a of the reflexive, symmetric relation Â2 U ^ , ( / i , / 2 ) LM,(/ 2 ,/,) on ^ 2 (cf. Exercise O 3.3D). 2.3B. Verify that the projection A2 -* Coker(/i —/ji) is t n e coequalizer of

2.3C.

2.3D.

2JE. 23F.

a parallel pair of morphisms AX^XA2 in the category Mod? of fl abelian groups. Let t be an endomorphism of a finitely generated free module V over a commutative ring K with basis /? : Z„ -» K Show that the K[T]homomorphism R(ß):K[T]" -* ^ is the coequalizer of the parallel pair (T, R( ß)tR( ß)'1) of endomorphisms of K[T]n. (a) Let A be an object of a category D with initial object _L . Show that the coequalizer of the pair ± =J A is the identity on A. (b) For a set X = {.*!,..., JC„}, show that the free group XG on .Xe s tne has the presentation {xx *„I)GP> '- - * coequalizer of — 0 G =? ZG. Determine the group with presentation {xx\x\ = 1 ) ^ . Determine the group with presentation (xvx2\x\* = 1, xxxx2xx =

2.3G. Let / be the path category of the directed graph ■ =J • or ({1,2} -» {1}, {1,2} -» {2}). Show that the identity functor on / does not have a coequalizer or equalizer. 2J H . (a) Show that the coequalizer of an equal pair of arrows in a category is the identity on their codomain. (b) Formulate and do the dual of (a). 231. Write out formal definitions of the categories DJ/A and DA/A involved in the formulation of Proposition 2.3.5. Then give a careful proof of the proposition.

234

CHAPTER III CATEGORIES AND LATTICES

23J.

Let X, Y, fx and f2 be as in Example 2.3.3. (a) For a given pair (
=

9(<7, T)'

(c) Determine the automorphism group Aut S3 of S3. (d) Does 5 3 have any outer automorphisms? 2.3K. (a) Let 0 -» E ->A1-+A2 be an exact sequence of abelian groups. Show that e : E -* A1 is the equalizer of the pair (/, 0) of ModTmorphisms. /i

(b) Let Ax^tA2

be a parallel pair of arrows in Mod7- Show that

their equalizer is the injection Kerv/i - f2) ~~* Av 2.3L. (a) Verify Proposition 2.3.6. (b) Formulate and prove the dual of Proposition 2.3.6. 2.3M. Let / be the poset category with Hasse diagram 1 -» 0 «- 2. Let D be a category. Let A : / -* D and B : J09 -» D be functors. (a) Show that the pullback of JA in D is characterized as a terminal object IT of the category DA/^4 of constant functors (from / to D) over A. (b) Show that the pushout of / o p B in D is characterized as an initial object i of the category B/DA of constant functors under B. 2 JN. Let C be a category with terminal object T. (a) Show that the product Ax xA2 of two objects A¡ of C is the pullback IT-,

Ax XA2

- A,

77-,

■*T

Ax(b) Show that the equalizer e : E arrows in C is the pullback

Ax of a parallel pair Ax ^XA2 of fi

-+AX

e A,

h

- A,

LIMITS AND LATTICES

235

(c) Identify coproducts as pushouts. (d) Identify coequalizers as pushouts. 2.30. Let C be a small category. Describe the domain of the composition of morphisms in C as a pullback C1 x C(j C,. 2.4. Groups in Categories In I 1.3, a group (5, fi, J, e) was defined as a set S equipped with an associative multiplication ¡x : S2 -* S; (s, t) •-> st, an identity element e satisfying (Vs G S, se = s = es), and an inversion J:S-*S; s <-> s - 1 satisfying (Vs e 5, ss _1 = e = J - 1 S ) . As discussed in Section 1.3, the element e of 5 may be described by the map e:T -» S from the terminal object T = {<»} to S with image {e}. Since an empty product is terminal [Example 2.1.2(c)], one may also describe the element or map e as e : S° -* S. Given e : 5° -» S, it is convenient to denote the composite S -> S° -* S by e¡. The associative law or semigroup property of 5 is then specified by the commuting diagram „, (M.1) „, (2.4.1)

M

(l.M) M

-> 5

in the category Set. The existence of an identity element, i.e. the monoid property of 5, is specified by the commuting diagram

s2

(u) s i ^

sl

(2.4.2)

in the category Set. Finally, the inversion or group property of 5 is specified by the commuting diagram

(1.-0 „,2 U.i) *s ¿ (2.4.3)

in Set. In other words, a group is a set S equipped with maps /u.: S 2 -♦ S, e:S° -» 5 and J : S -> S such that the diagrams (2.4.1)-(2.4.3) commute.

236

CHAPTER III CATEGORIES AND LATTICES

Note that the commutative law for a multiplication fi:S2^>S can be expressed, using the isomorphism T:S2-> S2; (SV S2) ~* ($2. S I ) °f Exercise 2A, as the commuting of the diagram -* S2 (2.4.4)

M

V

in Set. Moreover, given groups S, and S2, a function f:Sl-*S2is& homomorphism if the diagram S2

(vif^zf)

(2.4.5)

grouP

*S¡ M

5,

/

-* S,

in Set commutes. The only special feature of the category Set of sets used in formulating these properties diagrammatically is the fact that an object S of Set has the powers S0, S1, S2, S3. The observation leads to the following: Definition 2.4.1. Let C be a category in which each object S has the powers 5° (i.e. terminal object of C), S1 = S, S2 = S X 5, and S 3 = S2 x S = S XS2. (a) A semigroup in the category C is an object S of C, equipped with a C-morphism /x : S 2 -» S (called multiplication) such that the diagram (2.4.1) in C commutes. (b) A monoid in the category C is a semigroup (5, /A) in C, equipped with a C-morphism e : 5° -» 5 and composite e, : 5 -» S° -* 5, such that the diagram (2.4.2) in C commutes. (c) A group in the category C is a monoid (5, /it, e) in C, equipped with a C-morphism / : 5 -» S (called inversion) such that the diagram (2.4.3) commutes. (d) A morphism of semigroups in the category C is a C-morphism / : Sj -> S 2 , with domain and codomain both semigroups in C, such that the diagram (2.4.5) commutes. (e) A semigroup (S, ¿t) in C is commutative if the diagram (2.4.4) commutes. D

LIMITS AND LATTICES

237

Example 2.4.2 (Sequences of Groups). Consider the poset category (N, ^ ) , and the functor category SetN. Note that SetN has pointwise finite products of objects (cf. Exercises 1.4F and 2N). Then a group in the functor category SetN is a functor A : N -* Set with object part n <-* An and with morphism part given by (n -* n + 1) ►»- (An -*An+1\ such that each An is a group and each /„ is a group homomorphism. Such an object is called a sequence of groups. Note that a finite sequence (II 1.1) of abelian groups and homomorphisms yields a sequence in the current sense on setting An = 0 for n > N. D

Example 2.4.3 (Groups in the Category of Groups). If (A, + ,0) is an abelian group, then + : A2 -» A is a group homomorphism, so that (A, + ) is a group in the category of groups. Conversely, suppose that the group (G, • , _ 1 ,1) is a group (G, p., /, e) in the category of groups. Thus Va, b, c, d e G, (oc, 6rf)/t = (a, ¿ V • (c, d)/x. Setting (a, ft, c, d) = (a, 1,1, d), one obtains (a, d)/x = (a, l)ji • (1, d)ju.. Setting d = e, one obtains a = (a, e)fi = (a, l)(i ■ (1, e)ft = (a, 1V • 1 = (a, l)/¿. Similarly, d = (1, d)¿¿. Thus (a, d)/¿ = a • d, and then acbd = ab-cd. Setting a = d = 1 yields eft = be, i.e. the group G is abelian. Thus groups in the category of groups are precisely the abelian groups. D Let G be a group, construed as a category G with G0 = {1} and Gx = G according to Example 1.2. For a unital ring S, an S-representation of G is a functor R : G -» Mody. The 5-module IR, the unique element of the image of the object part of R, is called a G-module, or the ipoce of the representation R. (Such representations are sometimes described as linear, in order to distinguish them from the representations G -» AH or G-sets X.) Let M be a G-module, with a representation (having morphism part) R:G -» Aut(A/, — ); g -» (m -» mg). It will be shown that M determines an abelian group p : E -» G in the slice category Gp/G of groups over G. The domain E of the group homomorphism p is called the split extension G[M or G[RMofM by G via the representation R. Its underlying set is G X A/, with multiplication (2.4.6)

( g i . m i ) ( g 2 l » i 2 ) = (8ig2>mig2

+

m2).

identity element (1,0), and inversion (g, m) -» (g~\- mg _1 ) (Exercise 2.4D). The group morphism p is the projection p : E -* G; (g, m) -» g. By Proposition 2.3.6, the direct square of p in the slice category Gp/G is E X G J? -» G; ((g. wji),(g,/n 2 ))->g. Define + G : £ x G £ -> £; ((gTm,),(g, w 2 )) -► (g,mj + m 2 ). Define 0G : G -» £; g-> (g, 0). Define a negation - G : £ - > E;(g,m) -* (g, - m ) . Then (p:E-* G, + G , - G , 0 G ) is an abelian group in the slice category Gp/G (Exercise 2.4E). Example 2.4.4 (Dihedral Groups). Let G be the two-element group = {1, /}, and let M be the abelian group Zn. Consider the representation

238

CHAPTER III CATEGORIES AND LATTICES

R : -» AutZ„; / -> (x -» -x). Then the split extension of Z„ by via R is the dihedral group Dn. D Example 2.4.5 (Direct Products). Let M be an abelian group. Let G be a group, and let R:G -» Aut M be the trivial representation g -» lM. Then the split extension of M by G via R is the direct product G X M. D Example 2.4.6 (Affine Groups). Let K be a commutative, unital ring, and let n be a positive integer. Consider the general linear group GL„(AD of dimension n over K (cf. Exercise II 2.4K), the group of invertible elements of the monoid (Knn, • ,/„). The matrix product K'l X K% -> K? (II 1.3.4) yields a ^-representation R:GLn(.K) -» Aut AT;1. The split extension of (#1", + ) by GL„(K) via R is called the general affine group GAn(K) of dimension n over AT. The split extension of (K[',+) by SL„(Ä:) via the restriction of R to SLn(K) is called the special affine group of dimension n over AT. D Given the split extension p : G[M -* G, one may recover the module M as Ker p = {(1, m)\m G A/}. Note that (Ker p)2 is a subgroup of the pullback (G[M) xc(G[M), and the addition on Kerp is the restriction of + G to this subgroup. Moreover, (1, m)T((g, 0)) = (g _ 1 , OXl, mXg, 0) = ( g " \ "iXg, 0) = (1, /ng). In other words, the action of G on M is recovered as conjugation of G0 C on Ker p. More generally, noting that (1, m)T((g, n)) = (g~\- ng~lX\,mXg,n) = (g - 1 , -ng _ 1 Xg, mg + n) = (l.mg + n), one obtains an action (2.4.7)

(g, n) : m -» mg + n

of the split extension G[A/ on M. EXERCISES 2.4A. (a) Verify the claims of Example 2.4.2. (b) Let A and B be sequences of groups. Show that a natural transformation a : A -» ß is a morphism of groups in SetN iff each component an : An -» £„ is a group homomorphism. 2.4B. Show that monoids in the category of monoids are commutative. 2.4C. Let an object (M, • ,1) of Mon be a monoid (Af, ¡i, e) in the category Mon. Show that the image of the monoid homomorphism ex: M -* M

STD.

2.4D. Verify that (2.4.6) makes E a group. 2.4E. Verify that p : E -» G is an abelian group (£, + G , - G , 0G) in the slice category Gp/G. 2.4F. Verify the claim of Example 2.4.4. 2.4G. Verify the claim of Example 2.4.5.

LIMITS AND LATTICES

239

2.4H. Show that the symmetric group S3 is isomorphic to the affine group GA,(Z3). 2.41. Is the symmetric group 5 4 isomorphic to the affine group GA2(Z2)? Justify your answer. [Hint: Consider (2.4.7).] 2.4J. Show that GAn(K) is isomorphic to the subgroup of GLn+l(K) consisting of matrices whose first column is [1,0 0] r . 2.4K. Let G be a group, and let (p : E -* G, fi, J, e) be an abelian group in the slice category Gp/G. (a) Show that e : G -» E is an injective group homomorphism. (b) Show that (Ker p)2 is a subgroup of E, and that p: E xGE -* E restricts to a commutative group operation + .(Ker/?)2 -* Ker p. (c) Let r : Inn E -» Aut Ker p be the restriction of conjugations to automorphisms of the normal subgroup Ker p of E. Show that the composite R : G -* E -» Inn E -» Aut Ker p yields a linear representation of G. (d) Show that p: E -* G and G[Kerp -» G; (g, n) •-» g are isomorphic objects of Gp/G. 2.4L. Let G be a (not necessarily abelian) group, with automorphism group Aut G. Define a multiplication on the set (Aut G) x G by (a, gX ß, h) = (aß,gßh). Show that (Aut G) X G with this multiplication becomes a group (known as the holomorph Hol G of G). If G is abelian, show that Hoi G is the split extension (Aut G) [G of G by Aut G via the representation 1 : Aut G -* Aut G. 2.5. Limits Proposition 2.2.7, Proposition 2.3.4, and Exercise 2.3M(b) reveal a certain similarity amongst the concepts of coproduct, coequalizer, and pushout. Dually, Proposition 2.2.8, Proposition 2.3.5, and Exercise 2.3M(a) connect products, equalizers, and pullbacks. These similarities are formalized by the following : Definition 2.5.1. Let D be a category, and let / be a small category. (a) For an object B of D, the constant functor BA:J-*D has morphism partí« ^*j) ~* (B -$B). (b) For a D-morphism f:B -* B', the diagonal transformation fA : BA -» 5'A has component / : B -* B' at each object i of /. (c) The diagonal functor A : D -> DJ has object part A : B -> 5A and morphism part A : ( / : B -» B') -» (/A : BA -» 5'A). (d) For a functor F : / -* D, the category DA/F of constant functors over F is the subcategory of the slice category DJ/F whose objects are natural transformations

240

CHAPTER HI

CATEGORIES AND LATTICES

are of the form /A : (

F) -» (w : (hjnF)A -+ F). (f) For a functor F : / -» D, the category F/DA of constant functors under F is the subcategory of the slice category F/DJ whose objects are natural transformations

B'A. (g) For a functor F :J-*D, the colimit is an initial object 1/ or 1 : F -» (linjF)A of F/Z)A or, by abuse of notation, the object linjF of D. For objects i of 7, the components t, : ¿F -* UnjF are known as the insertions of the colimit. Given an object

: linjF -> ß is the D-morphism yielding the unique (F/DA)-morphism (ljig(p)A:(i:F-»(linjF)A)-*((p: F-> BA). D Remarks 2.5.2. (a) As a mnemonic for the direction of the arrow under the symbol "lim,"display a typical component with the functor value on the left : IT;

l¡

iF *~ UjnF and iF -* linjF. Note that the direction of the arrow under the symbol "lim"is the same as the direction of the component. "Projections are thrown out from limits : insertions are forced in to colimits." (b) Some authors use "lim" in place of "Jim "and "colim"in place of " lim. (c) Some authors use "projective limit" or "inverse limit" in place of "limit." Dually, they use "inductive limit" or "direct limit" in place of "colimit." Occasionally, these terms are used only in restricted circumstances, such as the "directed" case in which the small category / is a poset category having upper bounds (not necessarily least upper bounds) of finite subsets. D

Example 2.5J. (a) Let F : ( N , s ) - » ( R u {<»},<.) be a functor between the poset categories. Then the colimit linjF is the limit lim„_JnF) of the sequence (nF\n G M) of (extended) real numbers. The existence of the insertions in:nF~* HrgF says that linjF is an upper bound of the real numbers nF. If B is any such upper bound, the arrow (F -» (linjF)A) -» (F -* BA) in F/RA yields the diagonal transformation (ljigF)A -* BA in RA, i.e. the colimit arrow ljrgF -» B in R, i.e. the inequality linjF ^ B. Thus lirnF is the least upper bound of the monotonie sequence (nF\n e f^J). (b) Let K be a unital, commutative ring. Let KN = Set(N, K) have the pointwise X-module structure. Define an operation * on KN by / * g(n) = Zi+m-n f(Dg(m). Define 8 in KN by S(n) = if n = 1 then 1 else 0. Then

LIMITS AND LATTICES

241

(K^,*,S) is a commutative monoid making KN a Ä-algebra. Define a functor F : (IM,*) - M& by (n -♦ n - 1) -► (Ä:[r]/7"'Ä:[7'] K[T]/T"-XK[T]; p(T) + TnK[T] ~> p(T) + T"-lK[T]). Then the limit limF is the object K* together with the projections 7r„ : KN -» J t [ r ] / r " K l r ] ; / ~ m.- 0 7(i)r'] + r-Jflr]. The tf-algebra tfN is often denoted by J£[[r]], and called the algebra of power series over K in the indeterminate T. A function f:N->Kis then written as a formal power series El-ofiiyr'. The property Vn e N, /TT„ = E'^fiOP + T"K[T] is sometimes expressed as / = lim„_,xE".^0,/(í)7*,. (c) Let G be a group, and let (&< m(G), ç ) be the set of finite subsets of G, ordered by containment. Define F : â"< „(G) -» Gp_ with morphism part sending a containment X «-» y to the inclusion <.Y>~ of the subgroup of G generated by X in the subgroup generated by Y. Then G = Hin F, with insertions t^ : {X) <-> G. D As for the special cases discussed at the beginning of the section, the issue of uniqueness of general limit and colimit objects arises. Proposition 2.5.4. Let F: / -> D be a functor with small domain. (a) If the limit object ü_mF exists in D, then it is unique to within isomorphism inD. (b) / / the colimit object Um F exists in D, then it is unique to within isomorphism in D. Proof, (a) Suppose that ir : LA -» F and ir' : L'A -» F are both limits of F. Then they are both terminal objects of DA/F. By Proposition 1.3.4, there is a (DA/F)-isomorphism /A : (ir : LA -» F) -* (ir' : L'A -» F). This yields a diagonal isomorphism /A : LA -» L'A, and so a D-isomorphism / : L -» L'. (b) Dual to (a). D Pi

Pi

At the end of Section 2.3, the pullback ÖJ xjca2 of ö] ->**-a 2 in a category C was constructed as the equalizer a¡ Xxa2 ~* ax X a2 of the ""IPI

parallel pair ÖJ X a2 3 * of morphisms, i.e. of the parallel pair {trxpv ir2p2) of morphisms nf-ia, -* x between products. Thus if the category C has appropriate products and equalizers, it has pullbacks. This is an example of a general property of limits: They may be constructed by equalizers and products. Theorem 2.5.5. Let F : J -» D be a functor with small domain. Then JimF is the equalizer of the parallel pair iT\ue]iruduF,T\u<Ejirud) of D-morphisms Oj e j jF - > n „ e jud\F '• The limit exists if the products and equalizer exist.

242

CHAPTER III CATEGORIES AND LATTICES

Proof. For an object

V*

ß

-» kF

lim v

B

Km?

IT»

TIP

hjmF

jeJ0

(2.5.1)
iF «-

n vud.»F

ue/,

n«^

nyf

Here the central object, putatively labeled hjmF, is the equalizer of the bottom right and right lower morphisms in D. Set vf : = e7r, : JimF -» iF. Consider a /-morphism v:vd0-* vdv Since en u6 /«" H ¿ MF = en„6y-jru
JimF

hjnF-

w,vdx

- vd0F

w^F

(1.4.2) at v in /, is verified, showing that there is a natural transformation TTF :(JimF)A -» F. Now in order that the bottom left and top right-hand squares commute independently of the choice of i or k in J0, the diagonals from B to Yl¡ejt JF n a v e t o be Tljej
V«
* ud0F

(2.5.2)

B

-T--udxF

Vud\

It follows that n y e y o ^ n u e / i i r t t ( / ( ) M J and Thus ç>H<
LIMITS AND LATTICES

243

element Jim

A/F)-morphism (Jim : BA -» F ) -* (irF :(JimF)A -» F). Thus TTF : JimF -* F is indeed a terminal object of DA/F. D Corollary 2.5.6. Let F : J -» D be a functor with small domain. Then JimF is the coequalizer of the parallel pair CLuSjuFtuä, £„ 6 / i i B i o ) of D-morphisms Y.ueJud0F -» Eye y jF : The colimit exists if the coproducts and coequalizer exist. Proof. Dual to Theorem 2.5.5. D Example 2.5.7. (a) Consider how Theorem 2.5.5 builds the pullback A t XXA2 P\ Pi of Ax -* X «- A2 in the category Set. The bottom right-hand square of (2.5.1) produces the equalizer of the functions (irA pu irA p2): Ax X X X A2 ~* X x X and (vx, TTX) : Ax x X xA} -+ Xx X, i.e. the subset {(a,, x, a 2 )|a, px = x, a2p2 = x) of Ay X X X A2. (b) (The Ring of p-adic Integers.) For a prime p, consider the functor +1 F : ( N , Ï Ï ) -» Ring with morphism part (n:(n + 1) -> n) -> (l/p" l -* Z//?"Z; jf 4- /7,,+1Z -*x+ p"Z). The ring JimF is called the ring ZpZ of p-adic integers. Theorem 2.5.5 builds ZpZ as a subring of Tlne\(Z./p"Z). Writing elements of TlnmN(Z/p"Z) as sequences (xQ + Z, xx +pZ,x2 + p2Z,...), the subring is the equalizer of the identity homomorphism and the "shiff'Uo + Z, JCJ + pi, x2 + p2Z,...)-» (xj + Z, x2 + pZ, x3 + p2Z,...). In other words, it is the set of sequences (x0 + Z, *, + /?Z, JT2 + /> 2 Z,... ) with xn + l + p"Z = xn + p"Z for all n. D A category D is said to be (small) complete if each functor F : J ^ D with small domain has a limit in Z). Dually, D is (small) cocomplete if each such functor has a colimit in D. Theorem 2.5.5 shows that a category is complete if it has all products and equalizers. Corollary 2.5.6 shows that a category is cocomplete if it has all coproducts and coequalizers. For example, Set is complete and cocomplete. In general, a category is said to be (small) bicomplete if it is both complete and cocomplete. EXERCISES 2.5A. Let F : (N, £ ) - » (R U {-«},£) be a functor. Prove JimF = lim„_„«F. 2.5B. Verify the claims of Example 2.5.3(b). 2.5C. Prove (1 - r ) " 1 = l i m ^ . E ^ r « ' in K[[T]]. 2.5D. Verify the claims of Example 2.5.3(c).

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CHAPTER HI

CATEGORIES AND LATTICES

2.5E. Define F : (H, <.) -* Gp_ with morphism part (m -» n) -» (5 m «-» 5„). Prove 5«, = linjF. 2.5F. Define F : (&>< J.N), g) -» Gj) with morphism part (AT g Y) -> (AT! -» Y!). Prove S„ = ljmFT 2.5G. Let G be the group M, and let F be the corresponding functor of Example 2.5.3(c). How does this functor differ from the functor of Exercise 2.5F? 2.5H. Draw the analogue of (2.5.1) that describes the proof of Corollary 2.5.6. 2.51. Use Corollary 2.5.6 to construct the pushout of the diagram P\ Pi Ax *- X ->A2 in the category Mod? of abelian groups. 2.5J. For a prime p, show that the sequence (1,1,1,... ) generates a subring of Zpl isomorphic to Z. 2.5K. Show that the category Mod? of abelian groups is complete and cocomplete. 2.5L. Let p be a prime. Let Mod(/) denote the full subcategory of Mody consisting of those abelian groups (A, + ,0) with the property: 3/1 e N . V a e A , pna = 0. (a) Is each abelian p-group an object of Mod(/)? (b) Show that Mod*/1 has equalizers. (c) Show that Mod*/' has finite products. (d) Show that Mod/ is not complete. 2.5M. Let p be a prime. Let Cp» denote the set of complex numbers c with the property: 3n e f^J. cp" - 1. (a) Show that Cp- is a subgroup of C*. (b) Show that Cp°> is a Sylov p-subgroup of ({z e C|zz = 1}, •, 1). (c) Show that Cp<° is the colimit of the functor F : (N, < ) -» Mody with morphism part (n -» n + 1) -» (Z/p"Z -» Z/pn + 1Z; x+p"Z~px+pn+iZ). 2.5N. Let G be a group. Show that the category G of G-sets and G-homomorphisms is complete and cocomplete. 2.50. Let F : J -* D be a functor with small domain. (a) If / has a terminal object T, show that linjF = TF. (b) Formulate and do the dual of (a). 2.5P. Show that - 1 has a square root in the ring Z5Z of 5-adic integers. (Hint: Consider roots of T2 + 1 in {0}, Z/5Z, Z/5 2 Z ) 3. ADJOINT FUNCTORS Let B be a subgroup of a group A. Consider the category A of /1-sets and /1-homomorphisms, and the category B_ of 5-sets and ß-homomorphisms.

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245

The forgetful functor G :A_-» B_ has morphism part (k : (Y, A) -» (Y',/4)) •-» (A: : (Y, ß ) -» (Y',B)l One may also write G = j, ¿, identifying G with the concept of restriction introduced in Section I 2.3. In the other direction, induction f^ leads to a functor F:Bj-^A_ with object part (X, B) <-* (X, B)\ i = ((X X /4)/5, A). Thus for Z in g,, the ,4-set AF is the free A-set over the 5-set X. Definition of the morphism part of F, e.g. acting on a fi-morphism f:X'-*X or ß-homomorphism f:(X',B)^>(X,B), depends on the ß-homomorphisms %. : (X', B) -» ((X' X A)/B, B) and T^ : (X, B) -» ((X X /D/fi, ß) of (I 2.3.2). Note that there is a composite ß-homomorphism fr\x : (A", B) -» ((Z X >!)/£, ,4)1 ¿. By (I 2.3.3), there is then a unique A -homomorphism JT)¿ : (X', B)î ^ -* (X, B)l g with % ' 7 % = /%• The image fF : X'F -* XF of / : X' -* X under the morphism part of F is defined to be this y4-morphism or A -homomorphism fr¡x. It is straightforward to verify that F does indeed give a functor F.B_->A_ (Exercise 3A). The restriction functor G and induction functor F are closely related. Their relation is described as adjunction. More specifically, G is described as a right adjoint of F, and F is described as a left adjoint of G. The relationship may be summarized by the isomorphism A(XF,Y)

(3.1)

= B(X,YG)

obtaining for a given B-set X and A -set Y. More explicitly, the isomorphism is çX:A{XF,Y)-+B{X,YGy,g~rlxgG,

(3.2) with two-sided inverse (3.3)

{rfY*

:B{X,YG)

-* A{XF,Y);

h

~h,

the /1-homomorphism Ti being given by (I 2.3.3). In order to describe ( Y). One may thus replace (3.3) by (3.4)

{ç*yX:g(X,YG)

-* A(XF,Y);

h ~ hFeY.

Incidentally, if B is the trivial subgroup {1} of A, and Y is an ,4-set (Y, A), then « y : Y X A -> Y; (y, a) -» ya is just the action (I 1.7) of y4 on Y. Remark 3.1. Various mnemonics are available to help one distinguish a right adjoint from a left adjoint. In the isomorphism (3.1), putting the unfunctored objects Y and X next to the isomorphism sign =, the left adjoint F is the

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functor appearing on the left-hand side of the isomorphism, while the right adjoint G is the functor appearing on the right-hand side. Another mnemonic, usually reliable when dealing with concrete categories, is that right adjoints like 1$ tend to be "trivial," "forgetful," or "destructive," while left adjoints like T g tend to be "complicated," "free," or "constructive." [Note that Example 3.1.6(b) below presents one case where this mnemonic has to be reversed.] D The isomorphism (3.1) is natural. More explicitly, for each X in Bg, there is a natural isomorphism Set with morphism part (k : Y -* Y') -* (B(X, kG) : B(X, YG) -> g(X, Y'G); h -» hkG). Then
wx

-^B(X,YG)

d(XF,k)

(3.5)

A(XF,Y')

BCX.kG)

p-


g(X,Y'G)

This diagram in Set commutes. Indeed, for g e A(XF, Y), one has g Set with morphism part ( / : X' -» X) -» (5(A", YG) -* ¡j{X',YG); h -» fh). Then
.

'(g~gk))->{k~(h~hka))

,.

x-

{

■" >(g"fFg))-*(f~(h~fi))

'

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247

where the relevant functors are being described by their morphism parts. Note how the right adjoint G appears on the right-hand side of the composite hkG in (3.6), while the left adjoint F appears on the left-hand side of the composite fFg there. In a similar vein, the B-homomorphism rjx : (X, B) -+ (X, B)îg lg becomes natural. It is the component at the 5-object X of a natural transformation Tj:l f i -»FG from the identity functor on B_ to the composite FG:(H~* Ap* §). To demonstrate the naturality, one needs to verify the commuting of the diagram X' —=—* X'FG VX'

(3.7)

f FG

/

nx

-» XFG

for a 5-morphism / : X' -» X. But VrfFG = lx-FFG is described as the unit of the adjunction (3.1). Dually, the ¿-morphism eY : YGF -* Y becomes the component at the ^-object Y of a natural transformation e : GF -* 1¿ (Exercise 3C). This natural transformation is called the counit of the adjunction (3.1). A major issue is the uniqueness of the isomorphism (3.2) or the unit component TJX : X -* XFG at an object X of B. The issue is addressed by exhibiting r¡x as an initial object of a specially constructed category, and then appealing to Corollary 1.3.5(a). The specially constructed category is a more general version of the category of constant functors under a given functor [Definition 2.5.1(f)] in which the colimit of the functor is an initial object. Indeed, no new construction would be needed, were it not for the fact (exhibited in Example 1.2.5) that the image of a functor need not be a subcategory of its codomain. Definition 3.2. Let S : C -» D be a functor, and let x be an object of D. (a) The comma category (x, S) ofx over S has objects f:x^*cS that are £>-morphisms from x to the image cS of a C-object c under the functor S. An (x, 5)-morphism y or yS : ( / : x -» cS) -» (/' : x -* c'S) is a C-morphism y : c -» c' whose image yS under S yields a morphism yS : ( / : x -» cS) -»

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(/' : x -» c'S) of the slice category x/D. The composite of two (x, S)morphisms yS : (f : x -> cS) -» (/' : x -► c'S) and -y'5 : (/' : x -* c'S) -* (/" : A: -» c"S) is ( r r ' ) 5 : ( / : x -» cS) -* (/" : x ^ c"S). (b) The comma category (5, x) of S over x has objects f:cS -* x that are D-morphisms to x from the image cS of a C-object c under the functor 5. An (5, x)-morphism y or yS : ( / : cS -» JC) -* (/' : c'S -> JC) is a C-morphism y :c -» c' whose image yS under S yields a morphism yS : ( / : cS -» x) -» (/' : c'S -* x) of the slice category D/x. The composite of two (S, x)morphisms yS : ( / : cS -» x) -> (/' : c'5 -» x) and y'S : (/' : c'S -» x) -» (/" : c"5 -> x) is (yy')S : ( / : cS -» x) -» (/" : c"5 -» *). D Example 33 (Slice Categories). Let l c : C -» C be the identity functor on a category C. Let x be an object of C. Then the slice category C/x is (isomorphic with) the comma category (l c ,x), while x/C is (isomorphic with) the comma category (x, l c ) . D Example 3.4 (Constant Functors Under a Given Functor). For functor F : J -> D with small domain, the category F/DA is (isomorphic with) the comma category (F, A). D Using comma categories, one may now recognize components of the unit and counit of (3.1) as initial and terminal objects. Proposition 3.5. Consider the adjunction (3.1). (a) Let X be a B_-object. Then i)x : X -» XFG is an initial object of the comma category (X,G). (b) Let Y be an A_-object. Then eY : YGF -* Y is a terminal object of the comma category (F,Y). Proof, (a) Let h : X -» YG be an object of (X, G). Then for A( YG). For any such morphism hG : (r}x : X -» .XFG) -> (A : -X -► YG), one has h = r¡x-hG = tupf, whence Ä = M
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249

Corollary 3.7. One has the following identities : (a) (riFXFe) = 1; (b) (GVXeG) = 1. Proof, (a) Consider an object X of B. Then at X, (a) reduces to the commuting of the diagram

XF

XF = —

► XFGF

XF

in A [cf. (1.4.4) and (1.4.5)]. Since eXF is terminal in (F, XF), there is a unique 5-morphism h:X-*XFG such that 1XF = hF ■ eXF = h(
EXERCISES 3A. If B is a subgroup of a group A, verify that induction \g does give a functor F ■§_-* A. 3B. Verify that be the functor of Example 1.2.5. Show that the comma category (0, F) is not a subcategory of the slice category 0/D. 3H. Write out a detailed proof of Proposition 3.5(b). 31. Write out a detailed proof of Corollary 3.7(b). 3J. Let B be a subgroup of a monoid A that is not necessarily a group. Show that the forgetful functor G:A_-> B_ with morphism part (k : (Y, A) -> ( y \ A)) -» (/t : (7, B) - (Y'f 5)) has a left adjoint (cf. Exercise I 2.3E).

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3.1. Adjunctions The adjunction relationship (3.1) between the restriction and induction functors is the prototype for a phenomenon that is extremely common, throughout mathematics and particularly in algebra. Definition 3.1.1. Let A and B be categories. Let F: B -> A and G:A-*B be functors. Then F is a left adjoint of G, and G is arightadjoint of F, if there is a natural isomorphism (3.1.1)

A(XF,Y)sB(X,YG)

for each object X of B and each object Y of A. In other words, the isomorphism (3.1.1) or (3.1.2)


^B(X,YG)

is the component at Y of a natural isomorphism -* Z X y of Example 1.2.2, along with the functor GY ¡Set-» Set whose morphism part is ( / : X - X') -* (XY -► X'Y; g -» gf). Then (ÖT4.4) becomes a bijection
(b) Dual to (a). D

The natural transformation 17 is called the unit of the adjunction (3.1.1); the natural transformation e is called its counit. Note that in the codomain FG

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of the unit, the left adjoint F comes on the left and the right adjoint G comes on the right. Spotting adjunctions is greatly facilitated by the following result, a key illustration of the theme broached in Section 1.3 that mathematical constructs are specified as initial objects. Theorem 3.1.4. Let G: A -» B be a functor. Then G has a left adjoint if and only if, for each object X ofB, the comma category (X, G) has an initial object. Dually, G has a right adjoint iff (G, X) has a terminal object for each X in B0. Proof. The "only i f direction follows as in Proposition 3.5(a), via Proposition 3.1.3(a). The "if'direction builds the left adjoint F : B -> A of G : A -» B in the way that induction was construed as a functor starting from (I 2.3.2) and (I 2.3.3). Suppose that, for each object X of B, there is an initial object 7)x of (X,G). Now T)x is a 5-morphism whose codomain is the image XFG of an /1-object XF. One thus defines a function F : BQ -* A0; X -> XF which will become the object part of the budding left adjoint F : B -* A. Given a B-morphism / : X' -» X, the ,4-morphism fF : X'F -* XF is defined as the /1-morphism yielding the unique (X', G)-morphism fFG: (TIX. : X' -» X'FG) -» (f-qx : X' -► XFG). As in Exercise 3A, one may readily verify that F : B -» A is a functor. Moreover, the definition of the morphism part of F yields r)x-fFG = fr¡x, so the initial object r¡x : X -> XFG of (X, G) becomes the component at A- of a natural transformation 17 : 1B -* FG [cf. (3.7)]. Now for X in B0 and Y in AQ, define (3.1.3)

9*

: A(XF,Y)

- B(X,YG);

g~

G

Vxg

.

Fix Y, and consider the 5-morphism / : X' -» X. For an element g of A{XF,Y), one has gA{fF,Y)fFGgG = fvxgG = (rixgG)B(f, YG) = gq>ZB(f, YG) . Thus A(fF, Y) B(X,_G) follows as in (3.5). Finally, for an element h of B{X, YG), define ht(ix to be the unique element of A(XF,Y) yielding an (X,G)-morphism (h^)G :{T)X:X-+ XFG) -*{h:X~* YG). Then ?$ and t¡fx are mutually inverse, so that (3.1.3) is an isomorphism. D Example 3.1.5 (Free Monoids). Let U : Mon -» Set be the forgetful functor whose object part sends a monoid (M, • ,1) to its underlying set M. Consider an object A" of Set as an alphabet. The universality property (O 4.1.2) shows that the inclusion i)x : X ^ X*U of the alphabet as the set of words of length 1 in X* U is an initial object of the comma category (X, U). Theorem 3.1.4 then shows that X -* X* is the object part of a functor F : Set -» Mon, the free monoid functor, that is a left adjoint to the underlying~set functor U:Mon-» Set. D

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Example 3.1.6 (Colimits and Limits). Let F : J -» D be a functor with small codomain. (a) By definition, the colimit i:F -» (linjF)A is an initial object of the comma category (F, A) (cf. Example 3.4). Thus the diagonal functor A : D -* D1 is the right adjoint of a functor linj : D1 -* D with object part F -> Urn F. (b) Dually, the limit tr :(hjnF)A -» F is a terminal object of the comma category (A, F) (cf. Exercise 3.3F). Thus the diagonal functor A : D -» DJ is the left adjoint of a functor hjn : DJ -* D with object part F -» Jim F. In this latter case, the heuristic of Remark 3.1 needs to be reversed: The left adjoint A here is trivial (or at least repetitive), while it is the right adjoint that incorporates the construction of the limit object. D Theorem 3.1.4 characterizes an adjunction between functors F : B -* A and G : A -» B in terms of initial objects for (X, G) or terminal objects for (F, Y). One may also characterize adjunctions without explicitly mentioning objects or morphisms of the categories A or B. Proposition 3.1.7. Let F: B -» A and G: A -» B be functors. Then F is a left adjoint of G iff there are natural transformations 17: 1B -* FG and e : GF -* 1A satisfying (TJFXFB) = 1 and {Gr¡XeG) = 1. Proof. The "only if' direction follows as in Corollary 3.7. For the " i f direction, suppose that such natural transformations exist. For an object X of B and an object Y of A, define
EXERCISES 3.1A. In the "Currying" Example 3.1.2: (a) verify that G y is a functor; (b) verify that
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253

(d) determine the unit TJZ : Z -> (Z x Y)Y; (e) determine the counit ex : XY x 7 - » I 3.1B. In the proof of Theorem 3.1.4, verify that F : B -* A is indeed a functor. 3.1C. Determine the counit of the adjunction in Example 3.1.5. 3.1D. Determine the unit of the adjunction in Example 3.1.6(a), and the counit of the adjunction in Example 3.1.6(b). 3.1E. By analogy with Example 3.1.5, show that the construction (I 1.4.3) yields a free group functor G:Set-» Gp that is a left adjoint to the forgetful functor i/:Gp_-» Set. 3.1F. Let F : Gp_-> Mon be the forgetful functor that forgets inversion. Show that F "Eus a right adjoint G : Mon -» Gp whose object part M <-* M* sends a monoid to its group of units (cfTExercise I 1.3M). What is the counit of this adjunction? 3.1G. Give a detailed verification of »I'X'PY = ^B(X.YG) m t n e proof of Proposition 3.1.7. 3.1H. (a) In the adjunction (3.1.1), suppose that A has a terminal object T. Prove that TG is a terminal object of B. (b) Formulate and do the dual of (a). 3.11. Suppose that G : A -» B has left adjoints F : B -» A and F : B -» A. Show that there is a natural isomorphism t, : F -* F'. [Hint : Use Theorem 3.1.4 and Corollary 1.3.5(a).] 3.1J. Let CSgp be the category of commutative semigroups and homomorphisms. Let G : CSgp -» Set be the forgetful functor. (a) Show that the construction of free commutative semigroups over sets (Exercise I 1.5E) yields a left adjoint F : Set -* CSgp to G:CSgp-> Set. (b) Show that CSgp has a terminal object T. (c) Identity TGF as the semigroup of positive integers. (d) Identify TGFGF as the semigroup of integer partitions. (e) Identify TGFGFGF as the semigroup of Segre characteristics. Now let e be the counit of the adjunction. (f) Identify the component (GFe)T of the natural transformation GFe as the map determining the sum of an integer partition. (g) Identify the component (eGF)T of the natural transformation EGF as the map determining the length of an integer partition. (h) Describe the map determining the sum of a Segre characteristic in terms of T, G, F, and e. 3.1K. Let Fx : A0 -» Al be a left adjoint of Gx : Ax -» A0. Let F2 : Ax ~* A2 be a left adjoint of G2 : A2 -*AV Use Proposition 3.1.7 to show that FXF2 : A0 -* A2 is a left adjoint of G2Gl : A2 -» A0.

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CHAPTER III CATEGORIES AND LATTICES

3.2. Equivalence and Boolean Algebras Let F : B -» A be an isomorphism of categories, with two-sided inverse G : A -» B. Since 1B = FG and GF = lA, there is an adjunction between F and G whose unit is the identity transformation 1: 1B -» FG and whose counit is the identity transformation 1 : GF -» \A. In this adjunction, F is a left adjoint of G. But there is also an adjunction between G and F whose unit is the identity transformation 1:1^-» GF and whose counit is the identity transformation 1 : FG -* \B. In this adjunction, F is a right adjoint of G. It is rare for the unit and counit of an adjunction to be identity transformations. On the other hand, it often happens that the unit and counit are natural isomorphisms. An adjunction (F, G, r¡, e) is an equivalence if the unit and counit are natural isomorphisms. Example 3.2.1 (Finite Cardinals). Let Set* " be the full subcategory of Set consisting of finite sets. Let Card< " be the small full subcategory of Set or Set< " whose set of objects is the set {[0, n)\n e N] of finite down-sets of the poset (N, ^ ) . Let G:Card <M -* Set <w be the inclusion functor. For each finite set X, pick a bijection -qx : X -* [Q,\X\\ Since t)x bijects, it is an initial object of the comma category (X, G). By Theorem 3.1.4, there is then an adjunction (F, G, 17, e). Note that, for n e N , the counit has component £ {Q, n) = l[o, »)■ Thus the adjunction (F, G, T?, e) is an equivalence. D If (F, G, 1,1) is an equivalence, it was noted that (G, F, 1,1) is also an equivalence. This symmetry holds for general equivalences: Proposition 3.2.2. Let (F, G, 17, e) be an equivalence. Then (G, F, also an equivalence.

S~X,T)~1)

is

Proof. By Proposition 3.1.7, one has 1XF = (r}F)x(Fe)x = vxeXF and 1YG = (Grj)Y(eG)Y = i7yG£y for X an object of the domain of F and Y an object of the domain of G. Thus l y G = (ey) -1 7jy¿ = (eY})Gr]ÇQ = ( e - ' O y i G î T 1 ) * and 1XF = «^(ijjf) -1 = *Í>(TÍJ?) - 1 = Ufe"'), A is said to be dmse if each object of A is isomorphic in /I to the image of an object of B under F. Note that

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255

the functor G of Example 3.2.1 is dense: Each finite set X is isomorphic (via Tfr) to the image of the object [0, \X\) of Card< " under the inclusion functor G. By the definition of Card< ", the functor G is full. Moreover, the functor G is faithful, since it is the inclusion of a subcategory. These three properties of the equivalence G are entirely typicalTheorem 3.23. Let F:B-*A be a functor. Then the following three conditions are equivalent: (a) F is an equivalence, (b) There is a functor G : A -» B with natural isomorphisms rj : 1B -» FG and e : GF -* 1A; (c) F is full, faithful, and dense. Proof, (a) =» (b) is trivial. (b) =» (c): Suppose that (b) holds. Note that (b) is symmetrical in F and G. The properties for F listed under (c) will be verified in reverse order. (i) F dense: For an object Y of A, there is an ^-isomorphism eY : YGF -* Y of Y with the image of the object YG of B under F. (ii) F faithful: Consider 5-objects X' and X. The naturality (1.4.2) of 17 at an element / of B(X', X) yields / = Vx'fFGVxl- Thus r : A{X'F,XF) -» B{X', X); k -* T)rkGrixl » a retraction for F : B(X', X) -* A{X'F, XF), whence F is faithful. Similarly, G is faithful. (iii) F full: Consider 5-objects X' and X. Given k : X'F -» XF, the naturality (1.4.2) of TJ at kr : X' -» Z yields Jt rfC = VJfc'ifr = kG. Since G is faithful, one then has krF = k, so that F : B(X', X) -» ytfX'F, AF) surjects. (c) => (a): Suppose that (c) holds. It will be shown that F appears in an equivalence (G, F, e~x, i7_1). Given an object Y of /I, use the density of F to choose an object YG of ß with an A -isomorphism e^1 '■ Y -* YGF. This A -isomorphism becomes an initial object of the comma category (Y,F). Indeed, given an object h : Y -» XF of this comma category, the fullness and faithfulness of F yield a unique 5-morphism g : YG -» A!" such that gF = £y/i : YGF -* XF. Theorem 3.1.4 now shows that there is an adjunction (G, F, e~\ 6). By Proposition 3.1.7, one has 1 ^ = (Fe)x1(0F)x = ex\9% at each object X of B, i.e. 0¿ = eXF : ZFGF -» XF. Since F is full, there is a 5-morphism / : X -> XFG with fF = e ^ : AF -» AfFGF. Since F is faithful, the equations (dxf)F = 0%fF = e ^ f e ^ = l^ f G f and (f9x)F = fF6£ = e xFexF = IA^F imply 0^/ = 1XFG and /ö^ = 1^. Thus 8 is a natural isomorphism, and the adjunction (G, F, e~\6) becomes an equivalence. D The conditions of Theorem 3.2.3 are very useful for determining the equivalence of categories. In the following example, condition (c) of Theorem 3.2.3 is used.

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Example 3.2.4 (Elementary Linear Algebra). Let K be a field construed as the small category of matrices over K (Example 1.4). Consider the category Mody" of finite-dimensional vector spaces over K (cf. Exercise 1.4B). Define a functor F : K -» Mody" with object part m -* ©", XK and with morphism part given by II (1.3.5). By definition of Mod£ ", the functor F is dense. By the isomorphism K{@\_XK, ®^_XK) = K{U'i.lK,n'¡'.lK) s K¡", the functor F is full and faithful. Thus F satisfies condition (c) of Theorem 3.2.3. It follows that the category of matrices over the field K is equivalent to the category of finite-dimensional vector spaces over K. D A second, major application of Theorem 3.2.3, this time using condition (b), is to Boolean rings. A unital ring (/?,+,-, 1) is Boolean if (R, • ) is a semilattice (cf. Example II 1.3.4). Proposition 3.2.5. (a) A unital ring R is Boolean iff V* e R, x2 = x. (b) A Booleanringhas characteristic 2. (c) The only Boolean integral domain is the Galois field Z 2 . Proof. The "only if* part of (a) is trivial. Conversely, suppose that the multiplication in a unital ring R is idempotent. Then Vx, y e R, x + y = (x + y)2 = x2 + xy + yx + y2 = x + y + xy + yx, so \fx, y G R, xy + yx = 0. Taking x = y, one obtains 2x = x + x = x2 + x2 = 0. In particular, (b) holds. In the ring R, one has xy = — yx = yx, so the multiplication in R is also commutative, and R becomes Boolean. Towards (c), let K be a Boolean integral domain. Each element of K is a root of the quadratic polynomial T2-Te K[T]. Thus K = {0,1} s GF(2). D In a Boolean ring (/?, +, •, 1), define x U y == x + y + xy [cf. Exercise II 1.3A(a)] and x C\y •= xy. Then (/?, U , n ) is a lattice: The absorption laws (2.7) and (2.8) are readily verified. In fact, (/?, U , n , 0,1) is a bounded lattice (Exercise 3.2G). Now the distributive laws in the ring (R, +, ■, 1) yield the meet distributive law (3.2.1)

xn (y Uz) = (x C\y) U (x n z)

and the join distributive law (3.2.2)

JCU ( y Hz) = (x U y ) n (x U z)

in the lattice (/?, U , n ) (Exercise 3.2H). In general, a lattice (L, U , n ) satisfying (3.2.1) and (3.2.2) is said to be distributive. Each of (3.2.1) and (3.2.2) alone is actually equivalent to the distributivity of a lattice (L, U , n ) (Exercise 3.21). Let (L, U , n , 0,1) be a bounded lattice. Let x be an element of L. Then an element x' of L is a complement to x in L if ((a)

< 3 -">

jcn*' = 0

(») ,

w

. i

•

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257

Note that an element of a bounded lattice may have several complements. For example, the subgroup <(12X34)> of the Vierergruppe V (Exercise I 3.2E) has complements <(13X24)> and <(14X23)> in the bounded lattice Sb V of subgroups of V (cf. Exercise 2G). However, an element x of a bounded distributive lattice has at most one complement (Exercise 3.2J). If x is an element of a Boolean ring (R, +, ■, 1), then x' = 1 - x is a complement to x in (R, U , n , 0,1) (Exercise 3.2K). A bounded distributive lattice in which each element has a complement is called a Boolean algebra. Note that the power sets of sets are Boolean algebras. Proposition 3.2.6. Let f:R-*Sbea lattice homomorphism between Boolean algebras. Then fis a bounded lattice homomorphism iff Vx e R, x'f = xf'. Proof. If / is a bounded lattice homomorphism, and x is an element of R, then xf U x'f = ( i U x')f = 1/ = 1 and dually xf(~)x'f= 0. Thus x'f equals the unique complement xf' to xf in S. Conversely, x'f = xf' implies 0/ = (x n x')f = xf n x'f = xf n xf' = 0, and dually 1/ = 1, so that / is a bounded lattice homomorphism. D A lattice homomorphism between Boolean algebras satisfying the equivalent conditions of Proposition 3.2.6 is called a Boolean algebra homomorphism. Let BAlg be the category of Boolean algebras and homomorphisms. Let BRing be the category of Boolean rings and homomorphisms. Given a Boolean ring (/?, +.0, •, 1), let (R, +,0, • 1)F be the Boolean algebra (R, U , n ,'). Given a Boolean algebra (R, U , n ,'), let (R, U , n ,')G = (i?, +,0, •, 1) with x + y = (x n y') U (y n *') (cf. Example II 1.3.4), with xy =x ñy, with 0 = X D J : ' , and with 1 = x Ux' (the latter two equations for any x in R). Then J?G is a Boolean ring [Exercise 3.2L(a)]. Given a Boolean ring homomorphism / : R -> 5, define fF : RF -> SF to be f : RF ^> SF. Observe that / F is a Boolean algebra homomorphism [Exercise 3.2L(b)]. Given a Boolean algebra homomorphism g : R -* S, define gG : RG -» RS to be g : i?G -» 5G. Observe that gG is a Boolean ring homomorphism [Exercise 3.2L(c)]. One readily obtains: Proposition 3.2.7. The assignments F : BRing-» BAlg and G :BAlg-> BRing yield mutually inverse isomorphisms between the categories of Boolean rings and of Boolean algebras. O Now let BRing< " be the full subcategory of BRing consisting of finite Boolean rings. Let Set* " be as in Example 3.2.1. Define a functor Spec: BRing< " -» ( S e t < a W with object part R ~* Spec R and with morphism p i r t T / : R -» S) -» ( r 1 : Spec S -» Spec R; P ** r ' ( / ' ) ) (cf. Exercise II 1.4S). Given a set X, let Zf = Set(X, Z 2 ) be the Boolean ring of functions to 7L2 = {0,1}, with componentwise ring operations. Given a function f:X' -» X, define Set(/, Z 2 ): Set(*, Z 2 ) -» Set(A", 2 2 ); A -» fh. Note that Set(/, Z 2 ) is

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a unital ring homomorphism. One obtains a functor Set( , Z 2 ) : (Set* <")op -> BRing. There is then a natural isomorphism 17 whose component at a finite set X is the function t\x : X -* Spec Zf; x •-» {/: A" -» Z 2 |x/ = 0}. Note that % clearly injects. To show that it surjects, consider a prime ideal P of the finite Boolean ring Zf. Let b be the upper bound tP of the sublattice PF of ZfT7. Note b e 7> c Zx implies 3 * e A". jrf? = 0, for otherwise 6 = 1. Then P c {/: X -* Z2|x/ = 0}. By Proposition 3.2.5(c), the prime ideal P is maximal. Thus P = {/: X -> Z2\xf = 0}, so that 17^ also surjects. In the other direction, there is a natural isomorphism e whose component at a finite Boolean ring 5 is es : Tf" s -» 5; xY -» EfKSpec 5 - Y). Here elements of Zf1"0 5 are construed as characteristic functions XY °f subsets Y of Spec 5. The intersection fl(Spec S - Y) of the complement of such a subset is then an ideal of 5. Now each ideal / of 5 can be expressed as the intersection 7 = fl {7> e Spec S\I £ P].

(3.2.4)

Certainly / is contained in the right-hand side of (3.2.4). Conversely, let x be an element of the right-hand side. Suppose x € /. Now the ideal (x' + /X5/7) of S/I is contained in a maximal ideal of 5/7. This maximal ideal is of the form Q/I for 7 <; Q e Spec 5 (cf. Exercise II 1.4J). Now JC í (2, for otherwise 1 = Jt U x' = x + x' + xx' e Q, a contradiction. But x g (2 ^ 7 in turn contradicts the assumption that x was an element of the right-hand side of (3.2.4). Given (3.2.4), one obtains x -» #{/.6Spec S|x S
The functors Spec: BRing" " -» (Set/ <")°P and Set(_, Z 2 ) : -> BRing" a satisfy condition (b) of Theorem 3.2.3, so that and (Set<<ü)op are equivalent. Proposition 3.2.7 then shows that is equivalent to the category BAlg"" of finite Boolean algebras. D

EXERCISES 3.2A. Show that equivalence is an equivalence relation on any set of categories. (Hint: Use Exercise 3.1K.) 3.2B. Let G :Set-» Gp_ be the free group functor of Exercise 3.1E. Determine precisely"which of the three properties of Theorem 3.2.3(c) are possessed by G.

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3.2C. Let t/:Gp-» Set be the forgetful functor. Determine precisely which of the three properties of Theorem 3.2.3(c) are possessed by U. 3.2D. Give an example of a full, dense functor that is not faithful. 3.2E. A functor G : A -* B is a reflection if it is part of an adjunction (F, G,r¡, e)—also known as a reflection—in which the counit e is a natural isomorphism. Prove that G is a reflection iff it is full and faithful. 3.2F. In Example 3.2.4, replace the field K by an arbitrary commutative, unital ring 5. Which of the properties (c) of Theorem 3.2.3 does the corresponding functor F still possess? 3.2G. Let (/?, + , •, 1) be a Boolean ring. Verify that (R, U , n ,0,1) is a bounded lattice. 3.2H. Let (R, + , •, 1) be a Boolean ring. Verify that (R, U , n ) is a distributive lattice. 3.21. Let (L, U , n ) be a lattice. Show that the following three conditions are equivalent: (a) L satisfies (3.2.1); (b) L satisfies (3.2.2); (c) L is distributive. 3.2J. Prove that an element x of a bounded distributive lattice has at most one complement. 3.2K. Let x be an element of a Boolean ring R. Show that x' = 1 - x is a complement to x in the lattice (R, U , n , 0,1). 3.2L. (a) Given a Boolean algebra (R, U , n ,'), verify that RG is a Boolean ring. (b) Given a Boolean ring homomorphism /, verify that fF is a Boolean algebra homomorphism. (c) Given a Boolean algebra homomorphism g, verify that gG is a Boolean ring homomorphism. (d) Complete the proof of Proposition 3.2.7. 3.2M. Given a Boolean ring ( 5 , + , - , 1 ) , define ( 5 , + , - , l ) F to be the corresponding ring (S, ® , °,0) of Exercise II 1.3A. Given a Boolean ring homomorphism / : / ? - » S, define fF : RF -* SF to be f : RF -* SF. Show that F forms a functor F : BRing -» BRing that is an equivalence. 3.2N. Let R be a finite, non-trivial Boolean ring. (a) Show that log2|i?| is a positive integer n. (b) Show that the automorphism group of R is the symmetric group Sn. 3.20. Suppose that a unital ring R has the property Vx e R, x3 = x. Prove that R is commutative.

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3.2P. (Atomic Boolean algebras.) An atom of a bounded lattice (L, +,, 0,1) is an element covering 0. The bounded lattice L is atomic if, for each non-zero element x of L, the interval [0, JC]S contains an atom. (a) For any set B, show that the power set (2B, U , O ,') is an atomic Boolean algebra. (b) Let R be the intersection of all the Boolean subalgebras of the power set (2R, U , n ,') that contain {(-», r), [r, «01 r e R). Show that R has no atoms. (c) Exhibit a Boolean algebra that possesses an atom, but is not atomic. (d) Let x be an element of a Boolean ring 5. Prove: (x'S e Spec 5) <= x is an atom of S. (e) Show that the converse of (d) is true if S is finite. (f) Show that the converse of (d) is false if S is infinite. (g) For any Boolean algebra R, define At R == {a e ¿?|2 =|[0,a]|}, the set of atoms of R. Show that eR : R -* 2MR; x - {a e AtR\a <. x} is a Boolean algebra homomorphism. (h) Show that eR injects if R is atomic, (i) Show that eR surjects if the lattice (R, U , n ) is complete. 3.2R. For a set X, define &>*°(X) = &>< ¿X) \J3»< "(X), the set of finite or cofinite subsets of X (cf. Exercise 2F). Show that the Boolean algebra (&>*£(Z), U , n ,') is not isomorphic to the power set (2B, U , n , ' ) of any set. [Hint: B finite =» 2B finite, while B infinite =» 2B is uncountable, by Proposition 1.1.1. Show ^^(Z) is infinite and countable.] 33. Galois Connections and Galois Theory Let (A, ^ ) and (B, <,) be poset categories. An adjunction (5, R, r¡, e) between functors S : B -» A and R: A -* B is known as a Galois connection. The relationship (3.1.1) reduces to: (3.3.1)

V x e ß . V y eA,xs

<,y <*>x<,yR.

The existence of the unit -q and counit e reduces to: (3-3-2)

(a) WL, (b)

Vx e ß . j c £xSR; ., Vy
On the other hand, given functors S : B -» A and R : A -* B such that (3.3.2) holds, applying 5 to the B-morphism (3.3.2Xa) and R to the ^4-morphism

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(3.3.2Xb) yield xs <. xSRS and yRSR <, yR. Meanwhile, setting y = xs in (3.3.2XW yields xSRS <; xs; setting x = yR in (3.3.2Xa) yields yR <. yRSR. Thus: (3 3 3)

' "

((a) [(b)

Vx(=B,xs =xSRS; VyeA,yR=yRSR.

In particular, Proposition 3.1.7 then shows that S is left adjoint to R. Summarizing: Proposition 3.3.1. Let (A, <,) and (B, <,) be poset categories. Then a Junctor S.B -^ A is left adjoint to a functor R:A-* Biff (3.3.2) holds. D Example 33.2. (a) (Annihilators.) Let M be a module over a commutative ring K. Let B = (2M, c) and let A = (ld(K), 2). Define 5 : B -► A; X ~ {k e AT|Vm e X, mk = 0} and i? : A -» 5; / -* {m G M|VA: G /, mjfc = 0}. Since V ^ ç M . ^ ç * s / ! and VKK,I z IRS, Proposition 3.3.1 shows that the functors 5 and R yield a Galois connection. (b) (Images and inverse images.) Let f:D-* C be a function. Define 5 : (2D, ç ) - (2 e , ç); X -» Xf and /? : (2 e , ç ) - (2 D , ç); Y -» / ^ Y. Since Ay £ y *» Z Qf~lY, the functors J? and 5 yield a Galois connection. (c) (Polarities.) A relation between sets / and / is a subset a of I X J (cf. Exercise 1.1F). As in the special case of binary relations on a single set, it is convenient to use the notation xay for the membership (x, y) e a. Given a relation, a between / and /, define S:(2', ç ) -» (2J, 2);-X"-» {y G J\Vx G * , xay) and J? : (2', 2) -> (2', ç); y - {* G /|Vv s Y, xay). Now for J f ç / and Y ç /, one has Z s 2 y «• V* G * , Vy G Y, xay « A' ç YR. Thus /? and 5 yield a Galois connection, known as the polarity determined by the relation a. D In a Galois connection (3.3.1), elements of the subsets BS of A and AR of ß are described as closed. One obtains induced posets (BS, ^ ) and (AR, z). By (3.3.3), the restricted functors 5 : AR -» ÖS and R-.BS-+AR then yield category isomorphisms. This isomorphism between the posets of closed elements is called the Galois correspondence for the Galois connection (3.3.1). For an element x of B, the closed element xSR of B is called the closure of x. By Theorem 3.1.4 [cf. Proposition 3.5(a)], the closure of x is the product of the set of closed elements above x in B, i.e. (3.3.4)

xSR=

Tl{yR\xiyR}-

Dually, one has (3.3.5)

yRS = £ {x5Us ïS y}

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for y in A (Exercise 3.3C). The closed element yRS oî A is called the kernel of the element y of A. Given a Galois connection, the determination of the closed elements on each side is often a primary concern. Example 3 .3.3. (a) Let G be a group. Consider the polarity determined by the binary relation {(x,y) e G2\xy = yx). Then the closed elements are subgroups of G. (b) Let X b& a finite set, and let Zf be the Boolean ring of functions / : X -* Z 2 . Define a relation a on X X Zf by xaf *> xf = 0. Consider the polarity determined by a. Then the closed subsets of Zf are precisely the ideals of Zf. (c) If B is a complete lattice in the Galois connection (3.3.1), the Tarski Fixed-Point Theorem 2.1.8 shows that the set of fixed points of the composite functor SR:B~*B forms a complete lattice. But this set of fixed points is precisely the poset of closed elements of B. Similarly, if A is a complete lattice in a Galois connection (3.3.1), then its closed elements form a complete lattice. If both A and B are complete lattices, Exercise 2L then shows that the Galois correspondence yields lattice isomorphisms. D Let G be a group, and let (X, G) be a G-set. The subset {(x, T) e X x G\xT = x) of X x G used in the proof of Burnside's Lemma (Theorem I 3.1.2) is a relation between X and G that yields a polarity. The right adjoint is the fixed-point junctor F:(2G, D ) -» (2*, ç);H -» Fix(A",> e AfF, VT 6 TV, yf/T = yUTU~lU = yU =* yU e NF.

a

Galois connections and Galois correspondences arose initially in "Galois Theory." a particular case of the adjointness between fixed-point and stabilizer functors. Suppose that a field A contains a subfield K, making A a üC-algebra AK via the inclusion K •-» A. Let G be the Galois group of AK, the group MgK(A, A)* of /C-algebra automorphisms of AK. Consider the G-set {A, G). One wishes to find conditions under which each subgroup of G is closed and each field B with K <, B <, A is closed. An obvious first requirement is that K itself be a closed subfield of AK. To see that this need not necessarily happen, note that any Q-algebra automorphism U of RQ has

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the set [0,°°) of squares as a -subset. Then U:(M, ^ ) -• (R, <;) is a functor, since x ^ y =» y - x e [0, °°) => yU - xU = (y - x)t/ e [0, °°) => *t/ ^ yU. But V* e R, Vr e (-00, x) n Q, Vi e (x, ») O Q, r <. x <. s =» r <. xU ^ i. Thus x = xi/, and U is the identity. This shows that R is the only closed subfield of R Q , and in particular Q is not a closed subfield. For K to be a closed subfield of AK with Galois group G, it is necessary and sufficient that K = GF. Indeed, K « Y F for G 2 Y implies i ç G f ç y f = tf. Note that the faithfulness of {A, G) implies As = {1}, while AT5 = G by definition, and of course {1}F = A Identification of further closed elements is facilitated by a dual pair of inequalities. Proposition 3.3.5. If K <.B <.C <,Afor subfields B and C of A with dim CB finite, then \ Cs \ Bs | < dim CB. Proof. By induction on dim CB. If B = C, the result is trivial. If there is a field Dwith ß ^ D ^ C , t h e n | C s \ ß 5 | = | C 5 \ Z ) 5 | | Z ) i \ 5 s | ^ dimC D dim DB - dim CB, by induction and Proposition II 3.5.2. One may thus assume that C = B(t) for an element t of C. Let p(T) be the minimal polynomial of t in CB. Since C s \ 5 s - » ( x e C\p(x) = 0}; CSU -* tu is a well-defined injection, |C s \ Bs | ^ deg p(T) = dim CB. D Proposition 3J.6. If AlgK (A, A)* = G ^. H ^ J for subgroups H and J of G with \J\H\ finite, then dim J„F ^ | / \ H \. Proof. Let {1 = Ux, U2,..., UJ be a normalized right transversal to / in H. Suppose that there were an independent subset [alt..., an+1) of the //F-space JF. Consider the element [a¡Uj] of Ann+X as the coefficient matrix of a homogeneous system of n equations in n + 1 unknowns. By Gaussian elimination, [a¡U¡\ is reduced to a column-reduced matrix of column rank r :£ n. Thus the homogeneous system has a solution (xlt..., xr, 1,0 0) in A^ + i. Now if (VI s ¡ S r , i ¡ 6 HF) were true, the first equation of the homogeneous system would yield the contradiction xxax + ■■■ +xrar + ar+l = 0 to the supposed independence of the subset {ax,...,an+l} of the //F-space JF. Thus 31 ^ i ^ r. x¡ ÇÊ HF, whence (3.3.6)

3FG//.

3\<.i^r.

x¡-xY*0.

Applying the field automorphism V to the original homogeneous system, one obtains a new homogeneous system with solution (x\,..., x\, 1,0,..., 0) in A1+1. Consider the element T of S„ with f/jF"1 e JUjT or UjTV e JU¡ for 1 ^< j' £ n. Then the j'-th equation of the old system becomes the yT-th equation of the new system. In other words, the coefficient matrix of the new system is obtained from the old coefficient matrix, postmultiplying by the permutation matrix pT. Thus (xx,..., xvt, 1,0,...,0) is also a solution to the original system, whence the difference \xx -x\,...,xr-xvT,0,0,...,0) of the two solutions is again a solution. But by (3.3.6), the i-th equation of the column-reduced system yields the contradiction x¡ - x\ = 0. D

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Propositions 3.3.5 and 3.3.6 combine to yield the Fundamental Theorem of Galois Theory 3.3.7. Let K be a field. Let A be a field containing K, with dim AK finite. Let G be the Galois group of AK. Suppose that K is a closed subfield ofAK. Then \G\ = dim AK. Moreover, each subgroup of G is closed, and each field C with K <.C <,A is closed. In particular, the Galois correspondence yields a bijection between the set of subgroups H of G and the set of fields C between K and A, with dim CK = \CS \ G\ and \H\ = dim AHF. Proof For each intermediate field C, one has dim C | f = dim C^FF <.\CS \KS | <; dim CK <. dim CSKF, and also C <. CSF by (3.3.2Xa). Thus C = CSF is closed. Also \G\ = | As \Ks | = dim AK. Then for each subgroup H of G, one has IHFS I = | l f 5 \HFS\<, dim\FHP <.| 1 \ H \ = \H\ <.\HFS I, and also HFS 2 H by (3.3.2XW. Thus H = HFS is closed. D EXERCISES 3.3A. In the annihilator Example 3.3.2(a): (a) Verify the functoriality of R and 5; (b) Show that the closed elements of B are submodules of M. 3.3B. In the image Example 3.3.2(b): (a) Show that each element of 2C is closed; (b) Exhibit a function / such that 2° contains an element that is not closed. 33C. Verify (3.3.4) and (3.3.5). 33D. (a) Verify the claim of Example 3.3.3(a). (b) Setting G = S4 in Example 3.3.3(a), determine which subgroups are closed. 3.3E. Verify the claim of Example 3.3.3(b). 3JF. Show that the only Galois connections between antichains A and B are pairs of mutually inverse isomorphisms S : B -» A and R : A -* B. 3.3G. (Concepts.) Let / be a set of items, classified according to whether or not they possess various properties from a set /. Let a be the subset of / X / consisting of those pairs (i, j) such that item i has property ;'. Consider the polarity determined by a. A concept is an element (X,Y) of 2' x 2J with Xs = Y and X = YR, i.e. a pair of elements that correspond in the Galois correspondence. The set X of items is called the extent of the concept (X, Y), while the set Y of properties is called its intent. By Example 3.3.3(c), the concepts form a complete lattice. Consider the following 1987 classification of major west European countries. ("Major" means having at least 218 inhabitants, while "west

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European" means anywhere in Europe between the longitudes ± 10°.) Countries are identified as "small" (having less than 2 M inhabitants) or otherwise "large," as "coastal" or otherwise "inland," and as "monarchies" or otherwise "republics." (For Luxembourg, the term "monarchy" is considered to include the term "duchy.") Small? Large? Coastal? Inland? Monarchy Republic? 1 0 1 0 0 1 Austria 0 1 0 0 1 Belgium 1 0 1 0 1 0 1 Denmark 0 1 1 0 0 1 Eire 1 1 0 0 1 France 0 1 1 0 0 0 1 Germany 1 1 0 0 0 1 Italy 0 0 1 1 1 0 Luxembourg 0 1 1 0 1 0 Netherlands 0 0 1 1 1 0 Norway 0 0 1 1 1 Portugal 0 0 1 0 0 Spain 1 1 1 0 1 1 Switzerland 0 0 1 1 0 0 1 0 UK

3.3H. 3.31.

3.3J. 3.3K. 3.3L.

(a) Show that {France, Germany, Italy} forms the extent of a concept whose intent is {large, coastal, republic}. (b) Determine the concept whose extent is the closure of {Belgium, Denmark}. (c) Determine the concept whose intent is the kernel of {inland}. (a) Show that Q is a closed subfield of Q(2*)Q. (b) Show that Q is not a closed subfield of Q(2*)Q. Let A = Kit) be such that t has a separable minimal polynomial fiT) in K[T\ Prove that K is a closed subfield of AK. (Hint: Mimic the proof of Proposition 3.3.6.) Let q be a power of a prime number p. Prove that GF(/>) is a closed subfield of the GF(/?)-algebra GFiq). (Hint: Consider the Frobenius automorphism.) Determine all the automorphisms and the subfields of the field GF(28). (Symmetric functions.) Let AT be a field. (a) For a positive integer n, show that the commutative /C-algebra K[TU..., Tn] of Theorem II 3.7 is an integral domain. (b) Let A = KiTv ...,Tn)be the field of fractions of K[TU..., TJ. Show that there is an action of the symmetric group S„ on K(TU ...,Tn) given by Vt/ e Sn, VI <, i ^ n, T¡ U = Ti(J. (c) Show that Sn is a subgroup of the Galois group of AK.

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(d) Let 5 = Fix(A,Sn). Show that S is a subfield of A properly containing K. (e) Show that Sn is the Galois group of the 5-algebra A s . (f) Set n ? . , ( r - 7p = L^oi-lVajT"-' e S[T]. The elements CTJ ... T> for \<.j <,n. (g) Set £ = K(ax trn). Prove dim K(TX Tn)E <, n\. (h) Conclude K(ax,..., FR is the limit of FR: J -> B. Dually, R preserves the colimit of F if the existence of the colimit t : F -* (ljnj F)AA of F implies that J ? : FÄ -» (linj F)RA B is the colimit of FR:J-*B. The functor /? is continuous if it preserves all limits (of functors F : / -» ^4 with small domain), and cocontinuous if it preserves all colimits. Continuity of functors follows from adjointness: Theorem 3.4.1. (a) Right adjoints are continuous. (b) Left adjoints are cocontinuous. Proof, (a) Let R : A -» B be right adjoint to S : B -» A in the adjunction (S, R, T), e). Let F : J -* A have small domain, and let v : (hjn F)AA -* F be the limit of F. Given an object

SXFe)]A4 : (iSXFs)]Ä}Afl];.(irÄ),. = ^ [ h m ( ^ S X F s ) ] ^ ) * = T,b[(
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267

disjoint union is the coproduct of the free monoids over the uniands, becomes an instance of Theorem 3.4.1(b). The fact observed in Exercise 2M(a), that the underlying set of a product monoid is the product of the underlying sets of the factors, becomes an instance of Theorem 3.4.1(a). Exercise 2M(b) shows that right adjoints need not be cocontinuous. Dually, left adjoints need not be continuous. (b) Consider the currying adjointness of Example 3.1.2. The cocontinuity of the left adjoint FY, resulting by Theorem 3.4.1(b), yields the distributive law (Z, U Z 2 ) X Y = (Z, X Y) u(Z 2 X Y), and indeed the infinite version [E{Z, 11 e /}] x Y = HZ, x Y\i e /} for arbitrary index sets /. (c) For complete lattices (A, ^ ) and (B, ^ ) , suppose that a functor R : A ~* B is right adjoint to a functor 5 : B -» A. As observed in Example 3.3.3(c), the induced posets (BS, ^ ) and (AR, ^ ) of closed elements are themselves complete lattices, and the Galois correspondence yields a lattice isomorphism R: BS -* AR. Theorem 3.4.1(a) now shows that R preserves all products. (Contrast with Exercise 2.1G.) D Let ( L , + , - , 0 , 1 ) be a bounded lattice. For each element a of L, consider the left multiplication S(a) : L -* L; x <-* ax in the commutative, idempotent monoid (L, •, 1). Now x^y=*xy=x=>ax-ay = ax =» ax £ ay, so S(a) : L -» L is a functor. The bounded lattice L is a Heyting algebra if each of these functors S(a) \L-*L (for any a in L) has a right adjoint R(a) : L -» L;y -* a \ v. In other words, (3.4.1)

VÛ,X,VGL,

ax ¿y «* x £

a\y

by (3.3.1). If L is a Heyting algebra, then the left multiplications 5(a), being left adjoints, preserve the coproduct + in the category (L, ^ ) by Theorem 3.4.1(b), so that (3.2.1) holds in L. By Exercise 3.21, it follows that Heyting algebras are distributive lattices. Similarly, since the functors R(a) are right adjoints, they preserve the product • in the category (L, ¿ ) by Theorem 3.4.1(a), so that a\(bc) = (a\b)(a\c) for elements a,b,c of L. Note also a \ a = 1 for each element a of L. Example 3.43. (a) (Boolean Algebras.) Let (L, U , n ,') be a Boolean algebra. Define a \y — a" U y. Then a n x ^ y => x = (a' U a) n x = (ä n x) U (a n x) £ d U (a n x) ^ ä U y = a \y. Conversely, x <.a\y =* a C\x <.a C\(d U y) = (a C\a)u (a C\y) = a r\y <,y. Thus Boolean algebras are Heyting algebras. (b) (Bounded Chains.) Suppose that the bounded lattice (L, +, •, 0,1) has a chain as its partial order (L, <,). Define a \y ¡= it a <,y then 1 else y. Since (L, ^ ) is a chain, a - * £ y < = > ( a ^ y o r ; c ^ y ) « » ; t ; £ a \ y. Thus bounded chains are Heyting algebras. D In the Boolean algebra Example 3.4.3(a), one has d = a' U 0 = a \ 0. In a general Heyting algebra (L, +, •, \ ,0,1), one may thus define the pseudo-

268

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complement (3.4.2)

a' = a \ 0

for a e L . B y (3.4.1), d <. a \ 0 =» a ■ d <, 0, so that a ■ d = 0 [cf. (3.2.3Xa)]. On the other hand, in the bounded chain Example 3.4.3(b), one has 0' = 0 \ 0 = 1, but d = a \ 0 = 0 for 0 < a. Thus (3.2.3Xb) is not satisfied in general —hence the "pseudo" in (3.4.2). Now for elements x, y of any Heyting algebra (L, +, •, \ , 0,1), one has (3.4.3)

y


~xy =

0**x
Thus there is a Galois connection with ' :(L, ^ ) -* (L, ^ ) left adjoint to ' : (L, 2>) -» (L, <0. Each part of Theorem 3.4.1 then yields (3.4.4)

(a + b)' = a'b'

for any a, b in L. Proposition 3.4.4. (a) In any Heyting algebra (L, <>), the inducedposet (L, <,) forms a Boolean algebra. (b) A Heyting algebra L is a Boolean algebra iff the Galois connection (3.4.3) is a category equivalence. Proof, (a) Since " :(L, ^ ) -» (L, <,) is a functor, one has a-b <. a =» (a • f>y < a", and similarly (a • W <. b". Thus (a • bf <> d' ■ b". Conversely, (a • b) ■ (a ■ by = 0 =» b ■ [a ■ (a • W] ^ 0 => a • (a • by <; 6 \ 0 = b' = ¿"' = ¿>" \ 0 =»fc"• a ■ (a • by <, 0 =» ¿" • (a ■ by <, a \ 0 = a' = d" = a" \ 0 => a" • b" • (a • by <, 0 => d' ■ b" <. (a ■ by \ 0 = (a • bf. Thus (a • W = a" • ft", i.e. " : L -* L is a meet semilattice homomorphism, and its image L is a meet subsemilattice of L. For a, b in L, one has a <, d' <, (a + bf and b £b" <, (a + W, so that (a + ¿)" is an upper bound of {a, b} in L. For any such upper bound c' in L, one has a + b ^ c' => (a + 6)" ^ c"' = c'. Thus (a + bf is the least upper bound of {a, b} in L. In particular, (L', ^ ) is a join semilattice (L, U ) with d U ¿' = id + 67' for a, b in L. Since 1 = 0' G L' and 0 = 1' e L, the poset (L, ^ ) becomes a bounded lattice (L', U , •, 0,1). Now for a, ¿>, c in L', one has c • (a U b) = c" • (a + by = [c • (a + b)]' = [(c -a) + (c b)T = (c a) U (c ¿>), so that (L\ U , • ,0,1) is distributive. Finally, for c in L, one has c U c' = (c + c')" = (c' • c'7 = 0' = 1, the second equality holding by (3.4.4). Thus the pseudocomplement c' of c is a genuine complement, and (L', U , • ,0,1) becomes a Boolean algebra. (b) Note that (3.4.3) is an equivalence iff x" = x for all x in L. Then if L is a Boolean algebra, one has V j t e L , j t = x". Conversely, (Vx e L , x = JC") implies (L, +) = (L', U ), so that L is a Boolean algebra, by (a). D Since left adjoints tend to embody useful and significant mathematical constructions, it is helpful to have indications of their presence. In other words, when does a functor R : A -» B possess a left adjoint 5 : B -+ A? By

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Theorem 3.4.1(a), the functor R has to be continuous. For this necessary condition to have much force as part of a collection of sufficient conditions, the category A should be richly endowed with limits, and preferably be complete. One is led to the question: When does a continuous functor R : A -» B with (small-) complete domain have a left adjoint? If A and B are poset categories, the answer ("always") is given by the following: Theorem 3.4.5. (a) Let R: A -* B be a continuous functor from a complete lattice {A, <.) toa poset (B, ^ ) . Then R has a left adjoint S:B -* A. (b) Let S: B -» A be a cocontinuous functor from a complete lattice (B, <) to a poset (A, £ ). Then S has a right adjoint R: A -* B. Proof, (a) Define S : B -» A; x -» U{z e A | x z zR). Note xx <. x2 => {z<=A |jc, n{zeAfa £zR] <; n{ze/l[x 2 <,zR) => *f ^ xf, s o t n a t 5 is a functor. To verify its left adjointness to R, Proposition 3.3.1 will be used. Indeed, Vx e B,x £ U{zr \:z e A, x < zR) = [n{z \z e^4, x ¿ zR}]R = x 5Ä , the first equality holding by the continuity of R. Also, Vy e ^ , y R Ä = Il{z eA\yR <, zR) ^ y . Thus Ä and 5 satisfy (3.3.2), as required. (b) Is dual to (a). D A completely {meet) distributive lattice is a complete lattice (L, U , n ,0,1) such that (3.4.5)

Vx<=L,VJcL,xn

L{y|ye/} = E{*Hy|ye/}.

For example, any finite bounded distributive lattice is completely distributive. Each left multiplication functor S(x) : L -» L;y >-* x C\y of a completely distributive lattice is cocontinuous. Theorem 3.4.5(b) then shows that such lattices are actually Heyting algebras, known as complete Heyting algebras. Now if the underlying lattice of a Heyting algebra is complete, Theorem 3.4.1(b) shows that (3.4.5) is satisfied, so that the Heyting algebra is a complete Heyting algebra. For example, complete Boolean algebras fcf. Exercise 3.2P(i)], such as the power set of any set (cf. Example 2.1.6), are complete Heyting algebras. More generally, any complete sublattice T of a power set 2X, for which the inclusion is a cocontinuous bounded lattice homomorphism, is a complete Heyting algebra: (3.4.5) in (T, U , n , 0 , X) is inherited from (3.4.5) in (2* U , n , 0 , X). Such a sublattice T of 2X is called a topology on the set X, and the pair (X, T) is called a topological space. Elements of T are called open subsets ofX in the topology T. Example 3.4.6. (a) For any set X, the set {0, X} forms a topology, the indiscrete topology on X. (b) For any set X, the full power set 2X forms a topology, the discrete topology on X.

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(c) If (X, ¿ ) is a poset, then the set XD of down-sets of S (cf. Example 1.2.3) forms a topology, the Alexandrov topology on X. In particular, the poset {0 < 1} with the Alexandrov topology is called the Sierpinski space. (d) For any set X, the union of {0} and the set &< "(X) of cofinite subsets of X [cf. Exercise 2F(b)] forms a topology, the cofinite topology on X. (e) Let (X, M) be an M-set for a monoid M. Then the set T of Af-subsets of X is a topology on X. (f) Let {B, n , X) be a submonoid of (2*. n , X). Then {Y ç X | Vy e Y, 3Z e B. j e Z ç y ) is a topology on A', the topology spanned or generated by the base B. (g) Let S be a subset of 2X. Let «S>, n , X) be the submonoid of X (2 , n , A') generated by S. Then the topology spanned by the base <5> is called the topology spanned by the subbase S. For example, if V is a finite-dimensional real vector space, then the set {/""'(/•,«) \r e R , / e U(V, R)} of open half-spaces is a subbase for the Euclidean topology E on V. (h) Let Y be a subset of a topological space (X, T). Then Î7 = {Z ç Y | 3 G e T. Z = G n V ) is a topology on Y, the subspace topology on Y. D If T is a topology on a set X, consider Tc = {X - G \ G e T}, the set of complements of open sets. Such complements are called closed subsets ofX in the topology T. Note that (Tc, n , U , X, 0 ) is then a complete Heyting algebra. For example, the union of {X} with the set âa< a(X) of finite subsets of a set X [cf. Exercise 2F(a)] is the set of closed sets of the cofinite topology on X. If (X, T) and (Y, U) are topological spaces, consider a function / : X -* Y and the right adjoint / " ' :(2 y , ç ) ^ (2^, £); Z>-*f~xZ to the functor (2* ç ) -» (2y, ç); W -* Wf [cf. Example 3.3.2(b)]. The function / is said to be continuous if f1 restricts to a functor f1 :(U, ç ) -» (7\ ç ) . There is a large concrete category Top whose objects are topological spaces and whose morphisms are continuous functions. It is the domain of the forgetful functor G •Jag-* Set with morphism part ( / : (X, T) -» (Y, [ / ) ) - ♦ ( / : X -» Y). This forgetful functor G is left adjoint to the indiscrete topology functor I '■ S e t -» T o p with m o r p h i s m p a r t ( / : X -» Y ) -> ( / : (A", {0, A"}) -* (Y, {0, Y})). The forgetful functor G is right adjoint to the discrete topology functor D : Set-» Top with morphism part ( / : X -* Y) -»

(/:CY,2*)-»(Y,2 r )). EXERCISES 3.4A. Write out a proof of Theorem 3.4.1(b). 3.4B. (Creation of limits.) Let R : A -» ß be a functor, and let F : / -» A be a functor with small domain. The functor R is said to create the limit of F if the existence of the limit Fi? of FR implies the existence of the limit ir : ( hjn F)A^ -» F of F such that 7rÄ = eu. For

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a unital ring K, show that the forgetful functor U : Mody -» Set creates the limit of each functor F :J -* ModK. (Hint: By Theorem 2.5.5, it suffices to show that U creates products and equalizers.) 3.4C. Let R : A -* B be a functor, and let F : / -+ A be a functor with small domain. If FR has a limit, and R creates the limit of F, show that R preserves the limit of F. 3.4D. (a) Use Theorem 3.4.1(a) and Exercise 3.1F to show that the group of units of an arbitrary product of monoids is the product of the groups of units of the factors. (b) Determine the primary decomposition of the abelian group ( Z ^ , •, 1)*. (Hint: Use Proposition II 3.5.4.) 3.4E. Let / : D -* C be a function, and let {Y¡ \ i e /} be a set of subsets of the codomain C. Prove f~l n / e / Y = D , e / r l y 3.4F. Give a direct proof [i.e. independent of Theorem 3.4.1(a)] of the continuity of the right adjoint R: A -* B in a Galois connection (3.3.1). [Hint: You only have to prove that R preserves products, i.e. greatest lower bounds of (possibly empty) subsets of A] 3.4G. Let A be & cofinite subset of an infinite set X. Determine the pseudocomplement A of A in the cofinite topology on X [cf. Example 3.4.6(d)]. 3.4H. Let a be a relation between sets I and /. Consider the polarity determined by a [cf. Example 3.3.2(c)]. (a) Show that the closed subsets of 7, under the order induced from (27, ç), form a complete lattice L. (b) Show that the closed subsets of /, under the order induced from (2y, ç), form a complete lattice M. (c) Show that the inclusion functor (L, ç ) ■-» (27, ç ) is continuous. (d) Show that the inclusion functor (M, ç ) *-» (2J, ç ) is continuous. 3.41. Let \ : L X L -» L; (a, b) >~* a \ b be a binary operation on a bounded lattice (L, +, •, 0,1). Show that (L, +, •, \ , 0,1) is a Heyting algebra iff the following four identities hold for all a, b, c in L: (a) a \ a = 1; (b) a ■ (a \ b) = a ■ b; (c) b(a\b) = b; (d) a\(b-c) = (a\b)(a\c). 3.4J. Verify the claims of Example 3.4.6(e)-(h). 3.4K. For posets {X, <.) and (7, <.), let f:(X, ¿ ) -► (Y, <;) be a functor. Show that / : (A', AZ>) -> (Y, YD) is a continuous map between Alexandrov spaces [cf. Example 3.4.6(c)]. 3.4L. (The Zariski topology.) Let n be a positive integer, and let K be a field. Define a relation a between K" and K[Tu...,Tn] by U, JcJa/rtT",,..., r„) ~ /?(*],..., xn) = 0 [cf. 11(3.8)]. Subsets of K" closed under the polarity determined by a are called varieties. (a) For varieties Vx = Y/* and V2 = Y2Ä with Yv Y2 ç # [ r „ . . . , J J , show that F, U V2 = (Y,Y2)Ä, where Y,Y2 = {Plp2 \Pi e Y,}. [Hint:

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0 c „ . . . . xn) e Vx U V2 =» VJ e {1,2}, 3/>, e yf. />,(*„..., *„) * 0. Then (pi/> 2 X*i,..., xn) # 0.] (b) Show that arbitrary intersections of varieties are varieties. [Hint: Use Exercise 3.4H(c).] (c) Conclude that the set of varieties is the set of closed sets of a topology on K". This topology is called the Zariski topology. 3.4M. Identifying Rf with R \ show that SL2(R) is a variety. 3.4N. Let n be a positive integer, and let 1 : R" -» R" be the identity mapping. Show that 1 : (W, E) -» (R", Z) is a continuous function from R" with the Euclidean topology to R" with the Zariski topology. 3.40. Verify that G : Top -» S_et is right adjoint to the discrete topology functor D : Set-* Top. 3.4P. Verify that G : Top -> Set is left adjoint to the indiscrete topology functor / : Set-* Top. 3.4Q. (Completeness of Top.) (a) Show that the forgetful functor G : Top -> Set creates limits. (b) Show that the forgetful functor G : Top -» Set preserves limits. (c) Show that Top is complete. 3.4R. (Cocompleteness of Top.) (a) Dually to Exercise 3.4B, formulate a concept of "creation of colimits." Then show that the forgetful functor G :Top-» Set creates colimits. (b) Show that the forgetful functor G : Top-» Set preserves colimits. (c) Show that Top is cocomplete. 3.4S. Consider SL2(R) as a subspace of the space R4 with the Euclidean topology, so that SL2(R) has the subspace topology. Show that SL2(R) is a topological group, i.e. a group in the category Top. 3.5. Existence of Adjoints Theorem 3.4.5 gave a partial answer to the question of the existence of a left adjoint to a continuous functor R: A -* B with (small-)complete domain. Dealing with the case where A is a poset category, i.e. A0 a set and each morphism class A(x, v) having at most one element, it showed that a left adjoint always existed. The main result of this section, the Freyd Adjoint Functor Theorem 3.5.2, answers the question for the more general case where A is locally small, i.e. each morphism ciass A(x, y) is a set. In such cases, a left adjoint need not necessarily exist. Take A to be the poclass (Set0, D ) (cf. Example 1.1.4). Note that A is (small-)complete, having products given by unions and equalizers given by domains [cf. Exercise 2.3H(b)]. Consider the continuous functor R : A -» Set whose object part sends each set to the terminal object T of Set (i.e. a singleton). Suppose that R had a left adjoint 5 in an adjunctioiTTS, R, TJ, e). For each set Y, the counit

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component (eY : YRS -* Y) = (eY : TS -» Y) = (TS 2 Y) in A would then exhibit Y as a subset of TS. No set TS is that big! Theorem 3.1.4 reduced the existence of a left adjoint to the existence of certain initial objects (in comma categories). The first step towards the Adjoint Functor Theorem addresses the existence of initial objects in complete categories. Let D be a category. A set KQ of objects of D is said to dominate D if each object of D is the codomain of a D-morphism whose domain lies in K0. For example, if 5 is a non-trivial ring, then Max S dominates the poset category (Prop S, 2). Proposition 3.5.1. Let D be a (small-)complete, locally small category. Then D has an initial object iff it has a dominating set. Proof. If the category D has an initial object 1 , then D is dominated by the singleton {±}. Conversely, suppose that D has a dominating set KQ. Let K be the full subcategory of D whose object class is the set K0. Since D is locally small, the category K is small. Let F : K -> D be the inclusion of K in D. Since D is (small-)complete, the functor F has a limit ir : /A -» F with limit object /. It will be shown that / is initial in D. Each object k of K is the codomain of the identity D-morphism Kk: k -* k whose domain lies in K0. Extend the function K : K0 -» Dx; k <-* Kk to a function K : D0 -» D¡; x -» KX such that dorn KX e # 0 and cod KX = x for each object * of D. Then define À : D0 -» D¡; JC -> A^ such that A, = ir¡Kx for Kx:j-^ x. Let L : D -» D be the constant functor with morphism part ( / : x -* y) -* (1, : / -» /). It will first be shown that A is a natural transformation A : L -* 1D from L to the identity functor on D. Consider the following diagram:

The outer rectangle is the naturality diagram (1.4.2) for A at the D-morphism f:x-*y. Thus A,/ = ir¡Kxf'= irkKy = A, must be verified. The right hand rectangle is the pullback diagram (2.3.5) for the pair {ny,Kxf) of Dmorphisms. The pullback exists, since D is (small-)complete. Then irjKxf = TThKpejKxf= irhKpekKy = irkKy, as required. The second equality is the commuting of the pullback diagram (2.3.5), while the other two equalities hold by the naturality of ir.

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Given that A : L -» 1D is natural, it will now be shown that / is initial in D. For k in KQ, the naturality (1.4.2) of A at the £>-morphism irk : I -* k yields the commuting of /

/ Vu

(3.5.1)

Since K : DQ -» Dx extends the identity function (1.1) of the category K, the function A : D0 -» Dx extends the function TT : KQ -» Dv The commuting diagram (3.5.1) may thus be rewritten as

TT,

7Ti

showing that A,A : (TT : /A -» F) -» (IT : /A -* F) is a (/CA/F)-morphism. The terminality of ir : /A -* F in KA/F then yields A, = 1,. Now consider an object x of D. The set £>(/, x) is certainly non-empty, since it contains the component A, of A. To complete the proof that / is initial in D, it will be shown that Ax is the unique element of DU, x). Indeed, given any element / of D(l, x), the naturality (1.4.2) of A at f:l-*x yields the commuting diagram /

/ /

-» x Thus / = 1,/ = A,/ = \x, as required.

D

A functor R : A -* B is said to satisfy the solution set condition if, for each object x of B, the comma category (x, R) possesses a dominating set. (This dominating set is sometimes called a solution set for x with respect to R.) One may then formulate the Freyd Adjunct Functor Theorem 3.5.2. Let A be a (small-)complete, locally small category. If a continuous functor R: A -» B satisfies the solution set condition, then it is a right adjoint.

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Proof. Let x be an object of B. By Theorem 3.1.4, it suffices to exhibit an initial object of the comma category (x, R). By the solution set condition, the comma category (x, R) has a dominating set. Let G : (x, R) -» A be the faithful forgetful functor with morphism part (fR : (x -> yxR) -» (x -» y2R)) ** (f'-y\ ^ÏT)- Since G is faithful and A is locally small, it follows that (x, R) is locally small. By Proposition 3.5.1, it thus remains to show that (x, R) is (small-)complete. Let F : J -» {x, R) be a functor with small domain. Since A is complete, the composite functor FG : J -> A has a limit B. Now the functor F : J -» (*, Ä) may be construed as a natural transformation 9 : xAB -» FGi? whose component FG/Î) -» (wi? : vi?AB -> FGi?). Let (l:()¿m F be the natural transformation whose component at the object j of J is the (x, i?)-morphism G)Ä yielding a (5A B /FGi?)-morphism Um (t¡/GR)AB : (4>GR : zRAB -» FGi?) -* (CÚR: yRAB -* FGR). Now for each object j of /, one has /(hjn iffGRX of [(JC, i?)A/FK«A, ft) : The component of hjn i/» at each object y of / is the (x, i?)-morphism (Urn tl/G)R:(f:x -> zR) -» (Urn

^/î). D Example 3.5J. Let Af be a submonoid of a monoid Af. Let R:M-* N be the forgetful functor from the category M of A/-sets to the category "Ñ of Af-sets with morphism part (/ : (Ax, M) -» (/42, Ai)) -» ( / : (Av Nj -> (A2,N)). For example, if Af and A/ are groups, then R is the restriction functor i" of Section I 2.3. Let X be an A/-set. Recall the construction (Exercise I 1U) of the free A/-set XX M over the set X, together with the insertion rj : X -* X X M; x '-» (x, 1). Let i£0 be the set of composite maps k:X-l+XxM - ^ ( Z X M)aR for M-congruences a on ( * X M, A/) such that V* e * , V/i e Af, Cm, l)a(x, n). Note that )t : X -» ( * X A/)"/? is an N-morphism, since JTIA: = (xn, l)a = (*, n)a = (x, 1)«" = (x, \)an = j:/t« for x in .Y and n in Af. Now KQ is a dominating set for the comma category (X, R). Consider an object / : X -* YR of this category. As in Exercise I 1U,

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there is a unique Af-morphism f.XxM-^Y with TJ/ = /. The First Isomorphism Theorem for M-actions yields a commuting diagram XXM (3.5.2)

— - Y

nat ker / (XXM)k"f—j

XfM

in M. The right-hand arrow is the inclusion of the M-subset XfM = {xfm\x e Z , m e M ) o f Y, while 8 is an isomorphism. One then obtains a commuting diagram X—-

Tj(nat ker / )

, -^{XxMf'^R 6R

(3.5.3)

X

xff

XßfR

YR

in N_, the unlabeled arrows being inclusions. For x in X and n in TV, note (x, n)k"f = (JC, l)n ker / = (x, l) k "//i = xf{0-x)Rn = xfn{6-l)R = xn/ÎO"1)" = (*n, l) ker A so that the top arrow in the diagram is an element of K0 (and thus of N¿). The outer rectangle of (3.5.3) now shows that K0 dominates (X, R). Thus R satisfies the solution set condition. Moreover, the category M is complete and R : Af-> TV is continuous (Exercise 3.5D). The Freyd Adjoint Functor Theorem 3.5.2 then shows that R has a left adjoint 5 : W-> M. If N and M are groups, then S is the induction functor T^ of Section I 2.3. D EXERCISES 3.5A. (a) Let C be a small category. Show that the limit of the identity functor C -» C is the colimit of the empty functor 0 -» C. (Hint: Mimic the last paragraph of the proof of Proposition 3.5.1.) (b) Let D be a small, complete category. Show that D is equivalent to a complete lattice. [Hint: For each x in D0, show that x is an initial object of the full subcategory of D with object set Wë 1 W)rfi-1 3.5B. Verify the naturality of ÎÎ in the proof of the Freyd Adjunct Functor Theorem 3.5.2.

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3.5C. Let R : A -» B be a functor with small domain and codomain. (a) Verify that R satisfies the solution set condition. (b) Show that Theorem 3.4.5(a) is a special case of the Freyd Adjoint Functor Theorem 3.5.2. (c) Dualize the Freyd Adjoint Functor Theorem 3.5.2 to obtain a generalization of Theorem 3.4.5(b). 3.5D. Let N be a submonoid of a monoid M. (a) Show that the category M is complete. (Hint: Show that the forgetful functor t/: A/-> Set creates limits, in the sense of Exercise 3.4B. (b) Show that the forgetful functor R:M_-* N of Example 3.5.3 is continuous. 3.6. Tensor Products of Modules Throughout this section, let K be a commutative unital ring. For /^-modules Y and X, the set K(Y, X) = Mod^iV, X) of ÍT-module homomorphisms g : Y -» X itself forms a ^-module (cf. Exercise II 2. ID). For a fixed module Y, one thus has a functor RY : Mod^ -* ModK with morphism part ( / : X -> X') -» (K(Y, X) -» K(Y,X'); g^gflUU: Mod. - Set; (X, +, K) -» X is the forgetful functor, and GY : Set -> Set; X -> A^r is the right adjoint in the Currying adjunction of Example 3.1.2, then one has the commuting diagram Mod. (3.6.1)

R

»Set G

YU

Mod K—jj—> Set of functors. Since U is faithful and creates limits (cf. Exercise 3.4B), while the right adjoint GYU is continuous, it follows that Mody is locally small and complete, and that the functor RY is continuous. (For a direct verification of the continuity, cf. Exercise 3.6A.) The Freyd Adjoint Functor Theorem 3.5.2 will be used to produce a left adjoint SY to RY. The adjunction between SY and RY is the linear algebraic analogue of the Currying adjunction of Example 3.1.2. For the Freyd Adjoint Functor Theorem to apply, the functor RY : Mod f -> Mod y must satisfy the solution set condition. Fix a ÍC-module Z. Now an object g : Z -* K(Y, X) of the comma category (Z, RY) is an element of the subset K(Z, K(Y, X)) of Set(Z, XY) = Set(Z, XGY). The Currying adjunction provides a bijection
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bilinear function gip? : Z x Y -* X, an element of the ^-module K(Z, Y; X) in the notation of Section II 2.4. Speaking loosely, the comma category (Z,RY) "is" the category of bilinear functions on Z X Y. Let i : Z X y -> M be the insertion 11(2.3.4) of the set Z X Y into the free Ä-module M = LI Z x y Ä on the generating set Z xY.lt will be shown that the union U NsMK(Z, K(Y, M/N)), over all submodules N of M, dominates (Z, i? y ). Consider an element y
M/N

-» M

(3.6.2) Z <-

Im-y

The left-hand square is an instance of 11(2.3.4), while the right hand square results from the First Isomorphism Theorem. Thus TV = Ker y, while p is the projection, i is an isomorphism, and j is the inclusion. One then has the following (Z, i?y)-morphism: ( i p )
K(Y,

M/N) (iJ)Ry

(3.6.3) y
-

K(Y,X)

Note that y = iy = ipij bilinear in (3.6.2) implies that ip is bilinear, and thus that the top row of (3.6.3) is a well-defined X-homomorphism. The solution set condition is verified, so the Freyd Adjoint Functor Theorem 3.5.2 yields an adjunction (5 y , RY, rjY, eY) for each ZT-module Y. Definition 3.6.1. Let K be a commutative, unital ring, and let Z and Y be ÍC-modules. Then the tensor product Z ®KY or Z ® Y of Z and Y is the image ZSY of the module Z under the left adjoint SY to the functor RY : Mod». -> Mod.: X -» K(y, Z). D Combining the tensor product adjunction with the restriction of the Currying adjunction, one obtains isomorphisms (3.6.4)

K{Z ® Y, X) s K(Z, K{Y, X)) s K ( Z , y ; X)

for /C-modules .äf,y, Z. In particular, consider the identity 1Z9Y as an element of K(Z ®Y,Z® Y). Its image under (3.6.4) yields a bilinear

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function (3.6.5)

®:ZxY->Z®Y;(z,y)

The isomorphism Z XY =Yx (3.6.6)

^z®y.

Z yields the commutative law Z®Y^Y®Z.

The cocontinuity of the left adjoint SY [Theorem 3.4.1(b)] yields the distributive law (3.6.7)

Y® ] J Z ( = i el

i el

\ly®Zi

Example 3.6.2. For any /^-module Z, one has K(K ®K Z, X) s K(K, K(Z, X)) = K(Z, X). The uniqueness of adjoints yields an isomorphism K ®KZ = Z. Explicitly, the isomorphism is given as K ®KZ -* Z; k 9 z -» kz. D Example 3.6.3. Tensor products of free modules are free. Indeed, using (3.6.7) and Example 3.6.2, one obtains (LI, e ,K) 9 (LI jeJK) s U lxJK 9K KmUIXJK. D Example 3.6.4. Let L be a /C-algebra. Then the bilinear multiplication L X L -» L; (/j, l2) <-* IJ2 yields a linear ß : L 9 L -* L; /, ® /2 -» /i/ 2 . □ Example 3.6.5. Let / be an ideal of the ring K. Let X be a AT-module. Then there are mutually inverse isomorphisms X ® K/J -» X/XJ; x 9 (k + J) -* xk + XJ and Z/A7 -» Z ® AT//; jt + XJ ~» x ® 1. In particular, if 7' is a second ideal of K, one has AC//' ® K/J = # / ( / ' + / ) . D EXERCISES 3.6A. Verify directly that the functor RY :ModK -» Mody preserves both products and equalizers. Conclude that RY is continuous. 3.6B. Let A^d < j1 <, 3) be /T-modules. Verify the associative law (*i 9X2) 3.6C.

9X3 aXt ® (Z 2 ®Z 3 ).

Let F be a JC-module. Define ®" ,K inductively by ® ? ,F = F and ®I'L11/ = F® [«f.- 1 F]. Prove K"(V;K) = tf(®?mlV~K\ 3.6D. Verify the commuting of (3.6.3). 3.6E. Describe the inverse Z -» K ® Z of the isomorphism K ® Z -» Z; A: ® z -* kz of Example 3.6.2.

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3.6F. Let Z and Y be /C-modules. Elements of Z ® Y in the image of (3.6.5) are called primitive. Exhibit a ring K and X-modules Z and Y such that Z ® y has elements that are not primitive. (Hint: Take Z and Y to be vector spaces over a finite field K. Use Example 3.6.3 to compare \Z X Y\ and \Z ® y|.] 3.6G. Let L be a ^-algebra. Determine £-homomorphisms / and r such that the associative law in the monoid (L, •, 1) becomes the commuting of a diagram 3

® L —'—* L ®L

/-i

M

in Mod_K, using the notation of Exercise 3.6C. Prove (O 3.7). Verify the commutative law (3.6.6). Verify the claims of Example 3.6.5. Compute the tensor product of the abelian groups Z 6 and Zu. (Hint: Use Example 3.6.5.) 3.6L. Let L be a commutative Jf-algebra. Show that the forgetful functor ModL -» Mod y has a left adjoint whose object part is of the form X <-* X ®XL. (If X is a real vector space, then X ®RC is known as the complexification of X.) 3.6M. Let A be a splitting field of a separable polynomial f(T) in K[T], for a field K. Prove A ®KA s U?l\fA. 3.6N. (a) For positive integers m and n, prove iC" ® K" = K"". (b) For [XiS]mXm in K£ and [yk,]HXlt in * ; , prove that [x,¡l hkiLxn i" *m ® K« corresponds to [xyywlmB> in X^i under the isomorphism of (a). / g 3.60. Let 0 -> A —> B —► C -> 0 be an exact sequence of abelian groups

3.6H. 3.61. 3.6J. 3.6K.

Let G be an abelian group. Prove that the sequence A ® G -» 5 ® G - * C ® G - » 0 i s exact [cf. Exercise II 2.1B(a).]

IV UNIVERSAL ALGEBRA

1. SETS WITH OPERATIONS By this stage, readers should be acquainted with a wide range of algebraic structures: monoids, semilattices, monoid actions, groups, quasigroups, loops, right quasigroups, right loops, rings, fields, integral domains, modules, algebras, lattices, Boolean algebras, Heyting algebras, etc. In the study of these structures, certain concepts keep recurring: homomorphisms, products, substructures, congruences, the First Isomorphism Theorem, freeness properties, etc. The mathematician's response to recurring patterns is to seek an explanation at a more basic level. This is the origin of universal algebra. Thus the first task of universal algebra is to provide a framework for the study of the properties shared by the great majority of the kinds of algebra so far encountered. Once this framework is established, it may then be used to facilitate an approach to the various new kinds of algebra one may meet in the future. Universal algebra is a broad and powerful branch of mathematics. Such breadth and power have their price. However, there is at least a choice of modes of payment: a direct and concrete but cumbersome and formidable formalism on the one hand, or a clean and sophisticated but apparently rather abstract approach on the other. Choices like this are not uncommon in mathematics. For example, differential geometry may be studied either by classical index-ridden tensor analysis, or with a more modern, sheaf-theoretic approach. Although readers may have their own tastes and preferences amongst these alternative modes of study, it is best if one is able to master them both. Monoid actions provide a prototype for much of universal algebra. The approach taken in the first part of this chapter—sets with operations—generalizes the concept of an automaton presented in Section I 1.1 as a state space X together with a set A of elementary events. The construction of word algebras or absolutely free algebras in Section 1.3 corresponds to the generation of the free monoid A* of words in the alphabet A. Thus this first part restricts itself to the analogue of actions of free monoids. The next part, 281

282

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introducing the concept of a variety of algebras, presents the analogue of M-sets (X, M) for general monoids M, i.e. for quotients of free monoids. These two parts both take the direct, formalistic approach to universal algebra. The more sophisticated approach adopted in the third part, with the concept of an algebraic theory, corresponds to taking an M-set as a functor M -» Set, regarding the monoid as a category in the sense of Example III 1.2. Finally, the "monadic" approach of the fourth part corresponds to taking the action X X M -» X of an Af-set X as a component ex of the counit in the adjunction between the forgetful functor M-* Set; (X, M) <-* X and its left adjoint. This approach makes precise the extent to which universal algebra is modeled on monoid actions. 1.1. Operations and Types Let A be a set, and let n be a natural number. Then an n-ary operation on the set A is a function o> : A" -* A from the n-th direct power of A to A. For n = 0,1,2,3,4 respectively, the term "n-ary" becomes: nullary, unary, binary, ternary, quaternary (cf. Section I 3). A unary operation is just an element of AA, a function from the set to itself. Now the zeroth direct power A0 of any set A is a terminal object T of the category Set, i.e. a singleton set {<»}. A nullary operation o> or constant on a (necessarily non-empty) set A is thus a function it» : {°°} -* A, corresponding to the element oc<w of A (cf. Section III 1.3). One often identifies an element of a set A with the nullary operation A0 -* A selecting that element. For example, the identity element e of a group Q is selected by the nullary operation e : Q° -* Q; » -» e. Given an n-ary operation . Recalling that the direct power A" may be realized as the uniform code A" of length n in the alphabet A (cf. Exercise O 4.1B), one often writes a typical element (au a2,...,an) of the direct power A" as a word axa2 ■■■a„. The n-ary operation ta may then be described as w.A"^A; (alta2,.. .,a„) -» axa2 ...ancj. The form axa2 ...an-* a - b + c of (I 2.2.5) in an abelian group A. Notation with the operation preceding its arguments is called prefix notation. For example, the unary operation of negation in an abelian group A is usually written as - : A -» A; a -» -a. (Of course, one has to be careful not to confuse this with infix notation for the binary operation of subtraction.) Finally, the juxtaposition of two or more factors to denote their product (e.g. under a semigroup multiplication) might be called an instance of nofix

SETS WITH OPERATIONS

283

notation. Nofix notation is also used occasionally to tacitly denote (or rather: not to denote!) the nullary operation selecting the empty word in the free monoid (A*, •, 1) over an alphabet A. Example 1.1.1 ("Black Boxes"). A useful intuition about operations on sets may be gained by considering them as components in electrical circuits. Let the set A represent the set of potentials or voltages that may be present in any of the connecting wires of a circuit. Then an n-ary operation = if ax < r then a2 else 0. (c) (Batteries.) Consider a DC circuit with voltages in the range A = [0, N]. Then (the ungrounded terminal of) a battery of voltage V e [0, AH is represented by the nullary operation A0 -* A; to •-» V. (d) (Threshold Gates.) The voltages present in a neural network lie in the real interval A = [0, N], with 1 < N. A neuron firing when sufficiently stimulated is modeled in the neural network by a "threshold gate." This circuit component has a positive number n of inputs, and one output. Its "threshold" is a value r e [0, N]. The threshold gate corresponds to the n-ary operation co : A" -» A with a,... an « = if a, + ... + a„ < r then 0 else 1. D A group A is described as a structure (A, •, _1 ,1) with three operations, namely with a binary operation A2 -> A; (a, b) -» a ■ b of multiplication, a unary operation A -* A; a -» a - 1 of inversion, and a nullary operation A0 -»/!;<» -* 1 selecting the identity element. A right loop A is also described as a structure (A, •, / , 1) with three operations, namely a binary operation of multiplication, a binary operation A2 -* A; (a, b) -> a/b of right division, and a nullary operation selecting the identity element. In these terms, the key difference between the group and right-loop structures is that the second group operation is unary, while the second right-loop operation is

284

CHAPTER IV UNIVERSAL ALGEBRA

binary. Given an n-ary operation e il> of T is called the signature of T. For « e i l , the natural number tor is called the arity of the operator <w. D Given a finitary type T :il -* N, a r-algebra or an il-algebra (A, T) or (/4,il) o/ fype T or o/ signature (CDT\(O e Í1) is defined to be a set A equipped with an operation a> : A*" -* A corresponding to each operator or element ta of the domain Í1 of T. The type T is often described by (or identified with) its graph {(ta, tar) e íl x N\ta e Í1}. Thus the group (A,-, ~\1) is an algebra of type {(•,2),(_1,1),(1,0)}, while the right loop (A, ■ , / , \ ) is an algebra of type {(, 2), (/, 2), (1,0)} and signature <2,2,0>. Note that an abelian group (A, +, -,0) written additively may be considered as an algebra of type {( +, 2), ( - , 1), (0,0)} using the unary operation — of negation as an inversion, or as an algebra of type {( +, 2), ( - , 2), (0,0)} using the binary operation - of subtraction as a right division. An algebra of signature <2>, i.e. with a single binary operation, is called a magma. (The term "groupoid" is also used in this context, but is better reserved to denote a category in which each morphism is an isomorphism.) Example 1.1.3. (a) A dynamical system (X, T) is an algebra of type {(T, 1)}. (b) For a monoid M, an M-set (X, M) is an algebra of type M X {1}. (c) For a unital ring 5, an 5-module (X, +,S) is an algebra of type {( + ,2)}U(5X{1}). D Example 1.1.4. Let (AÍ, ■, 1) be a monoid. For elements 0 e l t . . . , xn) of M", define n^jJc, inductively by Il?_iJC, = 1 and I ! ^ , * , = (.Ur¡Zlx¡)xr for r <.n. Given a finitary type T:ÍI-*Ñ, one then obtains a r-algebra (M, il) with (1.1.1)

» : M - T - M ; ( * „ . . . , * „ ) - » El*, i-i

for each a> in il. D Consider a fixed set A. For a natural number n, the full set of all n-ary operations on A is Set(/T, A), described by the type function Set(^4", A) -* [n). Then the full set of all finitary operations on A is the disjoint union E neN Set(^4", A), described by the coproduct type function T or (1.1.2)

TA = ( £ (Stt(A",A)

^ {n}): £ Set(A",A)

-> N).

SETS WITH OPERATIONS

285

For a monoid M, an A/-set structure on A may be described as (A, M), or more precisely by means of the representation R: M -» Set (yl1, ^4) as in (I 1.1). In fact, this representation may be construed as a morphism R:(M -* {1}) -> (SetM1, A) -» {1}) of the slice category Set/{1}. Then for a finitary type T : il -* N, a r-algebra structure (^4,r)or(/l,il)on ^4 may be described more precisely by a representation of T on A, i.e. by a morphism R:T^>TA of the slice category Set/fcJ. This added precision is desirable when a set carries different structures of the same type. For example, consider the type r = {( ft, 2)} and a lattice (L, +, • ). Then the join semilattice structure (L,T) of L is defined by the representation J:T-* TL with ¡jj : L2 -* L;(x, v) ►»- x + y, while the meet semilattice structure (L, T) has representation M : T -» TL with ¡iM : L2 -» L; (JÍ, >>)►-* J: -y. Example 1.1.5. Consider the poset (Z + , ;g) of positive natural numbers with the usual ordering as a category C. Then the category's codomain function dx: Cj -> C0; (i ^ «) -» n is a finitary type _L : C, -* N; (i £ n) <-> n. Given a set A, one has a representation UA : ± -* TA with (i < n)ïlA defined as the n-ary projection onto the i-th factor IT" : A" -* A; (a, a¡,..., an) -» a,. The representation 11^ is called the projection representation. The image of n^ is denoted by ±A, or more explicitly as the type ±A : 0 -» f*J [cf. (1.3.7Xc)]. D EXERCISES LIA. Construe the digital stopwatch of Example 11.1.2 as an algebra of type {R,S,T)X{1). LIB. Determine a type for each of the following kinds of algebraic structure: semigroup, semilattice, quasigroup, non-unital ring, unital ring, tf-algebra (over a unital commutative ring K), lattice, bounded lattice, Boolean algebra, and Heyting algebra. 1.1C. Let A be a set. Show that (A, n2, v2) is an associative right quasigroup and that (A, IT2, v2) is an associative left quasigroup. LID. Let K be a commutative, unital ring, and let « be a positive integer. Define a type r : K[TX,..., T„] -» [n). Show that II (3.8) is a representation yielding a r-algebra (K, T). 1.2. The Isomorphism Theorems Throughout this section, let T : ii -» I^J be a finitary type. Given two algebras (A, fl) and ( 5 , Ü) of type T, a function / : A -* B is said to be an il-Aomomorphism or a r-homomorphism, or just a homomorphism, if (1.2.1)

a

\f-aUTfoi

= ax...am7taf

286

CHAPTER IV UNIVERSAL ALGEBRA

for all a) in ft and o, in A. Note that the identity 1A is an ft-homomorphism, and that the composite of ft-homomorphisms is again an ft-homomorphism. One obtains a large category T such that TJ, is the class of all algebras of type T, and T, is the class of all ft-homomorphisms. Note that the forgetful functor G:T-> Set, with morphism part (/:(/!, ft) -* (B,ft)) -+ {f: A -> B), is faithful. Example 1.2.1. Let T be the type {(-,2),(1,0)}. Then the concrete category Mon of monoids is a full subcategory of T. In particular, (1.2.1) reduces to 1/= 1 for the nullary operation 1. In general, (1.2.1) reduces to &>/= a» for a nullary operation w : A0 -* A; « -» &» on A and o> : B° -* B; °° -> CJ on B. D Let B be a subset of an algebra (A, ft) of type T. Definition 1.2.2. (a) Suppose Va> G ft, (VI <, i <, o)T, ¿j, G ß ) => ¿ ! . . . èWTa) e ß. Then the subset B is said to form a subalgebra (5, ft) of (A, ft)—notation (5, ft) ¿ U , ft) or 5 ^ ^ — w i t h (o-.B^^B; (bv...,¿>WT) -» è , . . . baTu for each a> in ft. One also says that {A, ft) is a superalgebra of ( 5 , ft). (b) Suppose Va) e ft, (VI ^ i < <úT,b¡eB) «=> è , . . . 6,,,Ta> G ß. Then the subalgebra (ß, ft) of (A, ft) is said to be a watf of (A, ft). (c) Suppose Va) G ft, Vfl!,..., amT G ^4, (31 ^ i <,
SETS WITH OPERATIONS

287

(b) The wallposet Vñ(A, il) or Wl A of (A, (I) is the subset of the power set 2A comprising the walls of (A, ft), with the partial order induced from (2A, ç). (c) The sink poset SkUA, ft) or Sk A of (A, ft) is the subset of the power set 2A comprising the sinks of (A, ft), with the partial order induced from 0.A, Ç ) . D If A is a set with an empty set 0 of operations, then Sb(y4,0) = Wl(y4,0) = Sk(A,0) = 2A. In the other direction, let ¥ be a subset of ft, and let a : W -* I^J be the corresponding restriction of T. Given an algebra (A, ft) of type T, one then obtains an algebra (A,^) of type o\ The algebra (A,V) is said to be a (basic) reduct or impoverishment of the algebra (A, ft). One also says that (A, ft) is an augment or enrichment of the algebra (A, ¥ ) . Note that there is a restriction functor /?* : r-> g, a faithful forgetful functor with morphism part (/:(y4,il)-» (B,ftJ)*->(f:(A,V) -> (B,V)). For the empty type 0 : 0 -> N, the category 0_ is just the category Set. Thus the underlying set functor U:r-* Set is an example of impoverishment. Note that reducís of a given algebra may have many more subalgebras than the original algebra. For example, f^J is a subalgebra of the reduct (Z, + ) of the group (Z, +, -, 0), but not a subgroup. Let (A, ft) and (B, ft) be algebras of type T. The direct product A X B then becomes an algebra (A X B, ft) of type T via (1.2.2)

a,:(AXB)mT^AxB;((al,bl),...,(amT,b„)) -»

(al...a„a,bi...b„a)

for each o> in O. In particular, direct powers Ar of A become algebras of type T for all positive integers r. Furthermore, A0 = T = {<»} is also an algebra of type T, with w = l r for all w in O. Now consider r-homomorphisms f:(C, ft) -* (A, ft) and g:(C,ft) -* (B,ft). The product map (f,g):C^A XB (cf. Proposition O 3.2.1) then becomes a r-homomorphism. Indeed, the diagram (O 3.2.4) becomes an instance of the product diagram III (2.3) in the category T. For any algebra (A, ft) of type T, the diagonal A [cf. (O 3.3.8)] forms a subalgebra of A2. In general, if (A, ft) and (B, ft) are r-algebras, then a relation a between A and B, i.e. a subset a of A XB [cf. Example III 3.3.2(c)], is called a T- relation (between A and B) if it is a subalgebra of A x 5. An equivalence relation a on ^4 is a congruence (relation) or r-congruence if it is a r-relation. In particular, a 0-congruence is just an equivalence relation. A r-algebra (A, ft) always has the improper T-congruence A2 and the trivial congruence A. If it has no proper, non-trivial congruences, then it is said to be simple. A simple algebra whose only improper subalgebras are empty or singletons is described as plain or strictly simple.

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CHAPTER IV UNIVERSAL ALGEBRA

Definition 1.2.5. Let (A, ft) be an algebra of type T. The congruence poset Cg(^4, ft) or Cg A of (A, ft) is the subset of the power set 2AXA comprising the congruence relations of (A, ft), with the partial order induced from (2AXA,c). U Recall (from Section I 3) that an n-ary relation a on a set A is a subset a of A". Such a relation is again called a r-relation (on A) if it is a subalgebra of the direct power algebra (An, ft). In particular, the subalgebras of the T-algebra A are precisely the unary r-relations. Proposition 1.2.6. Let (A, ft) be an algebra of type T. Then the posets Sb(/1, ft), WIM, ft), Sk(vl, ft), and CgC4, ft) form complete lattices. Proof. Let (L, ç ) denote any of the four posets under consideration. Then if / -» L; ;' <-* /, is a function, the product TljeJlj is the intersection If A' is a subset of a T-algebra (A, ft), then the subalgebra (X)sb(A,n), wall (X)wiM,O) o r sm^ (-^OskM.n) generated by X is the intersection of the respective set of all subalgebras, walls, or sinks of (A, ft) containing X. For a binary relation ß on A, one similarly defines the congruence < ß)c$A,a) generated by ß as the intersection of the set of all congruences of (A, ft) containing ß. If a is a congruence relation on (A, ft), then A" is a T-algebra with af.. .a%Ta> = a, ...a^tu™ for
n a t k e r a : ( ^ , f t ) - (Aa, ft); a ~» a a .

Conversely, one has the following: First Isomorphism Theorem 1.2.7. Let f : (A, ft) -» (B, ft) ¿>e a r-homomorphism. Then (1.2.4)

k e r / e Cg(/4,ft) and y4/e Sb(ß.ft).

One obtains the following commuting diagram in T : /I (1.2.5)

natker/ Ak"f

*B ; . r— /4/

//ere i w an isomorphism, and) : Af -* B is the inclusion of the image off.

SETS WITH OPERATIONS

289

Proof. By O 3.3.1, one already has (1.2.5) as a commuting diagram in Set. Now (1.2.4) is readily verified. The rest of the theorem follows by (1.2.3) and the definition of U k e r f, Cl). D There are two more classical isomorphism theorems to accompany the first. (Although the theorems are classical, their ordering varies.) Their proofs are left as Exercises 1.2E and 1.2F. Second Isomorphism Theorem 1.2.8. Let (B,il) be a subalgebra of a r-algebra (A, Cl). Then for a e CgiA, fi), one has: (a) L)Ba = {a e A\3b e B. aab) e SbU, fl); (W(UBa)ísfi("níI). D Let f:(A, Cl) -* (B, Cl) be a T-homomorphism. Define (1.2.6)

f*-CgB

-* CgA; / 3 ~ k e r ( / n a t 0 ) .

Note that / * : (Cg B, ç ) -» (Cg A, ç ) is functorial. Third Isomorphism Theorem 1.2.9. Let a be a congruence on a r-algebra (A, Cl). Then (nat a)* : CgM") -> Cg A induces an isomorphism (1.2.7)

( n a t a ) * : C g ( ^ a ) -> [a, A2]; ß*

(nat

a)*(ß)

between Cg(Aa) and the interval [a, A2] in the poset (Cg A, ç). Moreover, for ß e Cg(Aa), one has (1.2.8)

(Aa)ß

= A(n*

"nß\

EXERCISES 1.2A. Let {A, il) and (B, Cl) be T-algebras. Show that a function f:A-+B is a T-homomorphism iff its graph is a r-relation between A and B. 1.2B. Let ere C g U . O ) . (a) Show that a?... a°T
290

CHAPTER rv

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1.2F. Prove the Third Isomorphism Theorem. 1.2G. Let j : (B, Cl) -» (A, il) be the inclusion of a subalgebra B in A. For a G Cg(A, SI), prove j*(a) = atlB2. 1.2H. Realize Exercise II 1.4J as a special case of the Third Isomorphism Theorem. 1.21. (a) Show that a quasigroup (Q, •, /, \ ) has no proper walls. (b) Criticize the following argument: Consider the "arithmetic mean" quasigroup of Examples I 2.3 and 1.2.3. The closed unit interval forms a subalgebra. Since this subalgebra is a quasigroup, it has no proper walls. 1.2J. In the notation of Example 1.2.3, determine all the walls of the product ([0,1] X [0,1],»). (Hint: Draw a picture!) 1.2K. Determine all the sinks of the semigroup (Z4, • ). 1.2L. Show that there is a functor T-» Posop with morphism part ( / : A -» B) ~ (/* : Cg B -+ Cg A). 1.2M. Let A be a set. Define t : A2 -> A2; (a, b) -> (6, a) and k : (A2)2 -» A2; ((a, b), (c, d)) -» if b = c then (a, d) else (c, d). For each element (a, a) of the diagonal A, consider the nullary operation {a, a):T -* A2; oo -> (a, a) selecting (a, a). Show that the subalgebra poset Sb(y42, k, t, Â) is the poset CgM, 0 ) of equivalence relations on A. 1.2N. (a) Exhibit a plain quasigroup. (b) Exhibit a simple quasigroup that is not plain. 1.20. Let T : SI -* N be a type. Given a monoid (M, ■, 1), let (M, il) be the T-algebra of Example 1.1.4. (a) Show that there is a functor MonT : Mon -» T with morphism part (/: (M, •, 1) - (AT, •, D) - ( / : (M, i ! T ^ (7V\ O)). (b) If T = {(¿i, 2), (t, 0)}, show that MonT is the inclusion functor on Mon. (c) If ÎÎ is empty, show that MonT is the forgetful functor Mon-» Set. 1.2P. Let « be a positive integer, and let Í be a field. The quotient GL„(K)/#* is called the projective general linear group PGL„(/0 of dimension n over K. The subgroup K*SLn(K)/K* of PGL„(/0 is called the projective special linear group or linear fractional group PSL„(/0. (a) Prove PSLn(K)
SETS WITH OPERATIONS

(f) Prove \?SLn(q)\ = (q"'1 /d)n?:¿(qn est common divisor of n and q - 1.

291

-
1.3. Word Algebras Throughout this section, let T : ft -» N be a finitary type. An automaton with state space S and set E of elementary events is an algebra (S, E) of type E X {1}. It is also an £*-set (5, E*) for the free monoid E*, the monoid of words in the alphabet E. The "word algebras" or "absolutely free algebras" constructed in this section are the analogues of E* for the general finitary type T : ft -» N. Let A' be a set. Form the free monoid (X U ft)* of words in the alphabet X U ft. The free monoid becomes a r-algebra ((X U ft)*, ft) on defining (1.3.1)



for each w in ft. The (r-)word algebra or absolutelyfree(r-)algebra XÍI or (Xil, ft) over X is then defined to be the subalgebra of ((X U ft)*, ft) generated by X. The elements of Xil are called Sl-words or r-words in X, or sometimes just words. This justifies the term "word algebra." Example 1J.1. For the type E X {1} of the automaton (5, E), the word algebra XE = (XW\K e X, w e £*}. Thus if T = {°°} is a terminal object of Set, one has the bijection E* -» TE; w -» °°w. D The name "absolutely free r-algebra" arises because the injective function (1.3.2)

tfr : X -> XSIU; x -» x,

defined identifying X with its image under the insertion into the disjoint union X U ft, will become the component at X of the unit 17 in an adjunction (ft, U, TJ, e) between a functor ft : Set-* T with object part X -* Xil and the underlying set functor U:T-> Set. Indeed, by Theorem III 3.1.4, it suffices to show that (1.3.2) is an initial object in the comma category (X,U). Consider an object h : X -» AU of this comma category. Identifying functions with their graphs, one has a subset h of the product algebra Xil XA. Let / be the subalgebra of XSl X A generated by h. Then the T-relation / is a r-homomorphism / : XÜ, -» A (cf. Exercise 1.2A), and one has the commuting diagram Vx - ^ (1.3.3)

XÍIU

fU - AU

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CHAPTER IV UNIVERSAL ALGEBRA

in Set. Now / is the unique T-homomorphism yielding (1.3.3), so that t)x really is initial in (X,U), and one obtains the desired adjunction (ft, U, TJ, e). Note that the homomorphism / : XSl -» A may then be identified as the composite hneA. Consider the power set monoid (&>(X), U , 0 ) . As in Example 1.1.4, it may be realized as a r-algebra (&>(X\ft) with Yx...Yu>r-* {x} embedding X in &(X) as the set of singletons. The corresponding homomorphism sne^X) : Xil -*&(X) is called the argument map arg: Xíl -*&{X); w -+ arg(w). Elements of the set arg(w) are called arguments of the word w. For example, one has argixj... xaTcûi) = {xx x^}, and of course arg(x) = {x} for x in X. Note that the image of the argument map is contained in the subset 9>< „(X) of &>(X). From now on, it will often be convenient to write (1.3.4)

/>={*, <*2<

-}

for the ordered set of positive integers. (There are typographic advantages to writing xn instead of n.) Consider the function max ° arg : Pft -» N selecting the maximum of the (finite) argument set of an ft-word in P. (Note that one reverts to the usual symbols 0,1,2, ... for the elements of N.) Since the maximum of the empty set of natural numbers is 0, one has max ° arg
(1.3.5) Define (1.3.6)

:

ft

= epi max° arg

and T : ft -* N; (w,_n) -> n. Note that there is a (Set/M)-morphism j : (T : ft -» N) -» (T : ft -» N) with toj = (xx... xara>, N thus yields three types:

(1.3.7)

(a) (b)

1(c)

T

: ft -+ M; o> -» (OT;

T' : PCL -* M; w •-» max»arg w ; T

: ft -» N; (w,n) -* n.

The_type T' : Pft -» ñsj is called the type derived from T : ft -» M. The type ? : ft -» f^J is called the closure of the type T. The words in Pft are called

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293

derived operators or terms, while the elements of ft are called basic operators. The set ft will be described as a closed operator domain, and specifically as the closure of the operator domain ft. Example 13.2. For each operator domain ft, the closure ft contains the elements (x¡, n) for 1 <, i\ <, n. In parallel with the notation of Example 1.1.5, the element (x¡, n) is called the n-ary projection operator IT" onto the /-th factor. D The following simple result is worth noting. Proposition 1 J.3. There are nullary operators in ft iff there are nullary operators in ft. Indeed, r-1{0} = r_1{0}. In particular, r_1{0} = r' -1 {0}. Proof. One has (w, 0) e r_1{0} <=> (w, 0) e ft <=> max ° arg w <. 0 «* max » arg w = 0 <=> arg w = 0 <=> w e r~'{0}. □ _ Now consider an algebra (A, ft) of type T. Consider an element (w, n) of ft. Set X = {*„..., x„). Note arg(w) ç X, so w e Jffti/. Realize the direct power A" as Set(.Y. J4) : The element a of Set (X, A) corresponds to (x",...,x") in the Cartesian power A". Now the derived operator w is an element of XSIU. Then using the counit from the adjunction (ft, U, 17, e), one defines the n-ary operation (w, n)A or (w, n) on ^4 by (1.3.8)

xl...x°n(w,n)

= Wa™eAU.

This makes the algebra (.A, ft) of type r into an algebra (A, ft) of type_r or an algebra (A, Pft) £f type r'. Now U , Pft)/?;' = (A, ft), while U , ft)/?^, = (y4, Pft) and (A, SÍ)FCT = (/I, ft). The operations of PSÏ on A are called the derived operations or fe/wj operations of the r-algebra (^4, ft). A basic reduct of (A, ft) is called a reducf of (.4, ft). The set ft of operations on A is called the closed set of operations or the clone of the r-algebra (A, ft), and the closure of the set ft of operations on A. The operations from ft are called the clone operations of the r-algebra (A, ft). The set ft of operations on A is said to be closed or a clone if ft = ft. Example 1.3.4. (a) (Opposite Monoids.) Consider a monoid (A/, •, 1) as an algebra (A/, ft) of type (r : ft -► N) = {( n, 2), (1,0)}, with i : T -» M; {00} ~* 1 and ¿i : M2 -* M; (m, n) -» mn. Then the binary derived operation x2xx ¡xM on M is the multiplication x2xx ¿1 : M1 -* M; (m, n) -» nm = m° n of the opposite of the monoid (M, ■, 1) {cf. I (1.4)}. (b) Consider a quasigroup (ß, •, / , \ ) as an algebra of type r = {j8, S, y) X {2} with xtx2 ß = xxx2, xxx2S = xx/x2, and xxx2y = xx \x2. Then the ternary operation P of (I 2.2.5) is the derived operation xxx2x2ySx2x-i8ß.

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CHAPTER rv UNIVERSAL ALGEBRA

(c) (Convex Sets.) A real vector space V may be construed as an algebra (V, + ,R) of type (T : il -» N) = {( + ,2)} U (R X {1}). Using infix notation for addition and postfix notation for scalars, denote the subset o (JC,(1 - r) + x2r\r e (0,1)} of PÍÍ by I . [The notation originates on writing I for the closed unit interval [0,1] and I ' for its interior (0,1), the open unit interval.] Consider the reduct (V, D of (V, +, U). It is convenient to write the derived operations from I ' in the form r:V2-*V; (x, y) -» xyr = x(l — r) + yr. (Note that the quasigroup multiplication » on R in Example 1.2.3 is the derived operation 1/2.) Then the convex subsets of the vector space V are precisely the subalgebras of the reduct (V, 1° ) of (V, +, R). D One readily verifies the following (cf. Exercise 1.3J): Proposition 1.3.5. Let (A, ÍI) be an algebra of type T : il -» N. Then SbU, il) = SUA, il). D On the other hand, consider the type (T : Q -» N) = {(1/2,2)}, and the subalgebra ([0,1], il) of the algebra (R, °) of Examples I_2.3, 1.2.3(a), and 1.3.4(c). Then {0} is a wall of ([0,1], il), but not of ([0,1], il), since 0 = Olir2 e {0}, but 1 € {0}. Thus Wl ([0, l],fi) # Wl ([0,1], il). One of the themes of this chapter is that monoid actions provide a prototype for universal algebra. In the other direction, each algebra of a given type supports two monoid actions. Proposition 1.3.6. Let (A, it) be an algebra of type T. Then the set r~'{l} of unary clone operations and r' -1{1} of unary derived operations form submonoids of (AA, ■, \A). Proof. The identity \A is the unary derived operation (*,, 1). Now the unary clone operations are either of the form (a, 1) for a nullary derived operation a selecting the element a of A, or else of the form {x^u, 1) for a unary derived operation xxu. Taking a further nullary derived operation b and a further unary derived operation u, one has the following composition table: («.1)

(6,1)

(JC,I;,1)

(6,1) (6,1)

(<w,l) (x,uM)

Since av is again a nullary derived operation, and x^v a unary derived operation, the unary clone operations form a subsemigroup of AA, and in particular the unary derived operations form a subsemigroup of AA. □ Example 13.7. Consider a finite quasigroup Q as an algebra (Q, il) of type (T : il -» H) = {(/3,2)} U ( ß X {0}) with multiplication ß. Then T ' - 1 { 1 ) is the multiplication group Mlt(ß, ß). For example, let qx and q2 be elements

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295

of Q. Then R(qx) = x^ß, R(qi)R(q2) = xxqx ßq2 ß, R(qx)L(q2) = q2xxqxßß, etc. If Riq^'1 = R(qxY, then / K ^ ) - 1 = xxqxßqxß ...qx ß, a word of length 2r + 1 in />ft. D The device used in Example 1.3.7—adjoining a basic nullary operation a : A0 -* A; °° -» a corresponding to each element of the underlying set A of an algebra—is often useful. Given an algebra {A, ft) of type T : ft -» N, consider the corresponding algebra (A,A U ft) of type (1.3.9)

(TA : A U ft -» IM) = ( ^ -» {0}) U (T : ft -

N).

Then the operations of the clone A U SI are called polynomial operations of the algebra (A, SI). For example, realize a commutative, unital ring K as an algebra (K, +, • ) of signature (2,2>. Then for a polynomial p(70 e K[T], the £-algebra homomorphism II (3.5) yields a unary polynomial operation K-*K;k~* p(k)oi(K,+,). Definition 1.3.8. Let (A, SI) and (^4,*) be two algebra structures on a set A. _ _ (a) Then (A, SI) and (^4,¥) are (clonally) equivalent if (A, SI) = (/!,¥). (b) Then G4, Ü) and M , ¥ ) are ipofynomially) equivalent if M.ZTTÏÏ) = (A,A~UW). D Example 1.3.9. (a) Let R be a Boolean ring, and let RF be the corresponding Boolean algebra (cf. Proposition III 3.2.7). Construe R as an algebra (R, +.0, •, 1) of type (T : SI -* N) = ({0,1} X {0}) U ({ + ,•} x 2). Construe i?F as an algebra (R, U , n ,',0,1) of type (tr : ^ -* N) = ({0,1} X {0}) U {(', 1)} U ({U, n } X {2}). Then (R, SI) and (R,¥) are clonally equivalent. (b) Let Q be a group. Construe ß as an algebra (ß, -, _ 1 ,1) of type (T : ft-» M) = {(•,2),(~1,1),(1,0)}. Again, construe ß as a quasigroup (ß, ß, S, y) of type (
296

CHAPTER IV UNIVERSAL ALGEBRA

1.3D. Consider a quasigroup Q as an algebra (Q, ÍI) of type (T' : il -» M) = ({ß, S, y) X {2}) U ( ß X {0}), extending the type from Example 1.3.4(b). (a) Exhibit a quasigroup Q with Mit Q = T' - 1 {1}. (b) Exhibit a quasigroup Q for which Mit Q is a proper subgroup of r'-Hl}. 1.3E. Let {A, il) be a r-algebra with both ^4 and il countable. Prove that the number of polynomial operations of (A, fl) is countable. 1.3F. Construe the set S of Exercise HI 1.1C as the set of polynomial operations of an algebra (N,Cl) with a finite operator domain il. Then complete Exercise III 1.1C if you have not already done so. 1.3G. Let (A, il) be a r-algebra. Show that the clone operation on A determined by the n-ary projection operator v" of Example 1.3.2 is the n-ary projection operation IT" of Example 1.1.5. 13H. Let (A,0) be a set equipped with the empty set of operations. Prove that the clone of (A,0) is the set 0 = {IT/1 : A" -* A\\ <. i; ^ n e N} of projections of Example 1.1.5. 1.31. Let T : il -* N be a type. (a) Show that the poset category (N, £ ) has finite coproducts. (b) Realize N as a bounded join semilattice (N, max, 0). (c) Realize the monoid (N, max, 0) as a T-algebra (N, il) according to Example 1.1.4. (d) Show that the (graph of the) type T, the closure of T, is a subalgebra (T, il) of ((P u f l ) x N , il). (e) Show that the (graph of the) type T', the type derived from ?, is a subalgebra (T', il) of (T, il). 1.3J. Prove Proposition 1.3.5. 1.3K. Following Example 1.1.1, show how to interpret derived operations as circuits composed of components represented by basic operations. 1.3L. Let (R, +,0, •, 1) be a Boolean ring, with corresponding Boolean algebra (¿?, U , n , ' , 0 , l ) . (a) Show that (/?,+,-, 1) and (R, U , n ,', 1) are clonally equivalent. (b) Show that (R, +, •, 1) and (R, U , n , 1) are not clonally equivalent. (c) Show that (R, +, •, 1) and (R, U , n ,') are polynomially equivalent. 1.4. Universal Geometry Let A be a set. Recall that the full set E m e N Set(/l m , A) of all finitary operations on A is the domain of the type function T or TA of (1.1.2). It is convenient to identify this function with its graph. Now for a natural number

SETS WITH OPERATIONS

297

/, consider the power set ^(A1), the set of all /-ary relations on A (cf. Section I 3). In order to distinguish from the concept of arity for operations, these relations are sometimes described as having length I. One thus has the length function &>(A') -> {/}. Then the full set of all relations on A is the disjoint union Lie^iA1), described by the coproduct length function L or (1.4.1)

LA = ( E {*(A') - {/}): Z^(A')

- N).

It is also convenient to identify this function with its graph. Thus an element of the set LA is a pair (X, I), where X Q A1. A relation p of preservation is defined between TA and LA, namely (1.4.2)

(
Thus the operation , iff the subset X of A1 is actually a subalgebra of the direct power algebra (A1, w). In the terminology of Section 1.2, the relation X is preserved by {JA), ç ) and (â"(LA), 2). Thus for a type T : fl -» N of operations on A, the domain of TS is the set of relations preserved by each operation from il. For a restriction A: T -» N of the length function LA, the type \R is the largest type for which each element of T is a ( Ai?)-relation. Consider an element (X, 1) of (the graph of the restriction) À. Then X e SbiA1, XR) = Sb (A1, JÏÏ), by Proposition 1.3.5. Thus the types \R of operations on A that are closed in the polarity (5, R, 17, e) actually form clones or closed types in the sense of Section 1.3. Now the poset categories (&>(TA), ç ) and (&>(LA), 2 ) are complete lattices. As in Example III 3.3.3(c), the induced posets (â"(LA)R, ç ) and (&>(TA)S, 2 ) are complete lattices, isomorphic via the Galois correspondence. Note that for the type ±A of Example 1.1.5, one has ±A = LA. Thus the type ±A is the lower bound of the complete lattice (â"(LA)R, ç ) . Example 1.4.1 (The Euclidean Plane). Let A = U2V Let 02(U) = {/ e GL 2 (R) \ffT = 1}. Let R : 0 2 (R) -» Aut Uj be the representation making R, an 02(R)-module, with action given by the matrix product R, X 0 2 (R) -» Rj (II 1.3.4). Define the group E 2 of Euclidean motions as the split extension of (R,, + ) by 0 2 (R) via R. Thus E2 is a subgroup of the general affine group GA2(R) (cf. Example III 2.4.6). Define (1.4.3)

T = {{(a, b) G A2\(a - b){a - b)T = d}|o ¿ d G R } .

Thus the elements of Y are the binary relations on A consisting of all pairs

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CHAPTER IV UNIVERSAL ALGEBRA

(a, ft) of points a fixed distance d apart. Define À = T X {2}. The unary operations in \R are precisely the elements of E 2 . D Example 1.4.2 (Convex Sets). Let A = R. Define ÎÎ = {/l2 -> / l ; (a, ft) -+ (1 - r)a +rf>10 < r < l}. Thus the elements of SÏ are the binary operations of weighted mean (e.g. arithmetic mean for r = 1/2). Define T = SI X {2}. Then the domain of rS is precisely the set of convex subsets of the finite-dimensional real vector spaces Rm. D Klein's famous 1872 "Erlanger Programm" set out to characterize geometry as the study of the relations on a set A invariant under the action of a group G on A. Example 1.4.1 illustrates how the most classical of all geometries, the Euclidean plane, fits in to this programme. Euclidean motions are precisely the rigid motions—those preserving the distance relations (1.4.3)—and Euclidean geometry concerns itself with those concepts, such as cosines of angles and areas of triangles, that are invariant under the group E 2 of Euclidean motions. From the current perspective, it now appears that Klein's view of geometry is excessively limited by its restriction to groups, i.e. to bijective unary operations on the set A. The polarity (S, R, 17, e) determined by the preservation relation (1.4.2) offers a broader view of geometry: the study of the relations on a set A invariant under an arbitrary type T : il -* f^J of operations on A, not necessarily unary and not necessarily bijective. Geometry characterized in this way as part of universal algebra may be described as universal geometry. Example 1.4.2 illustrates how convex geometry fits in to this framework of universal geometry. Certain relations on the set A are invariant under each type T : (l -* N of operations on A To describe them, it is convenient to realize each direct power A1 as SetiZ,, A), the element a of Set(Z„ A) corresponding to (la la) in the Cartesian power A1. A function / : Z t -» Z, induces a function / * :Set(Z,, A) -» Set(Z t , A); a -» fa, and the corresponding direct image function

r-.^(A')

^(Ak);y~ry={fa\a<=y}.

An /-ary relation 8 is said to be diagonal if it is of the form f*A' for some function / : Zk -» Z,. For example, the binary equality relation  = {(a, a) I a e A) is diagonal in this sense, being / M 1 for / : Z 2 -» Z,; 1 -> 1,2 -> 1. The quaternary relation A X A is diagonal, being f*A2 for / : Z 4 -» Z 2 ; l - » l , 2 - » l , 3 - » 2 , 4 > - » 2 . Let A or A^ denote the full set of all diagonal relations on A, of all possible lengths. Let DA : A^ -» f^ be the restriction of LA to AA. Note that, if an operation
SETS WITH OPERATIONS

299

each power A1, it follows that the terminal object of the complete lattice (&>(TA)S, ç ) is DA. A type T : il -* N of operations on A, and the algebra (A, Û), are said to be functionally complete if ? = TA, or equivalently if TS = D^ (Exercise 1.4B). Example 1.4.3. Let (A, +,-,0,1) be a finite bounded lattice such that (A, rs) is a chain. Consider the type (T : ÎÎ -» N) = ({ + , } x {2}) U (A X {0}) u M x {1}). Construe A as a T-algebra, in which the nullary operation (a,0)A selects the element a of A, and where the unary operation (a, 1)A acts as (1.4.4)

{a,\)A:A

->A; x ~>x" = (if x = a then 1 elseO),

the "truth value" [* = a]. An arbitrary n-ary operation -* xx... xn_1 xn / o n A may then be expressed via x1...xn_lxnf=

(1.4.5)

f:(xu...,xn_1,xn)

E * , . . . * „ _ , « / • JCJ OS/4

in terms of il and (n - l)-ary operations. Since the full set Set (A0, A) of nullary operations on A is contained in il, one thus has / e fl by induction on n, so that the algebra (A, il) is functionally complete. D An algebra (A, il) is said to be polynomially complete if A with the clone i 4 U f l of polynomial operations is functionally complete. In other words, for each natural number m, each element / of Set(/l m , A) can be realized as a polynomial operation of the algebra (A, il). Example 1.4.4. Let (K, +, -, 0,1) be a finite field. For each element xm) of Km, define the delta polynomial

o-4-*)

«<„ *jTi,...,Tm)=

ft

n

j - \ x,+keK

^-|

Xj — K

in K[TV...,Tm] (cf. II (3.6)). For an element / of ^(Km,K), Lagrange interpolant polynomial (1-4-7)

(x¡

E

xm)eKm

{xl...xmf)S,Xi

(xu...,

define the

xJTu...,Tm).

Since the image of (1.4.7) under the .K-algebra homomorphism II (3.8) is /, it follows that the finite field K is polynomially complete. D

300

CHAPTER rV UNIVERSAL ALGEBRA

EXERCISES 1.4A. For a type T, prove TS - fS. 1.4B. For a type T, prove T = TA <=» TS = DA. 1.4C. Let G be a permutation group on a finite set A. Let T be the type G x {1}. Determine the set of unary operations of the type TSR. 1.4D. Show that a 2-element Boolean algebra is functionally complete. 1.4E. Let e be a primitive element of the field Z p of prime order p. Express the exponential function exp : 2p -» Zp; x -» ex [cf. (II 3.5.1)] as a unary polynomial operation of the ring Zp. 1.5. Clones and Relations The universal geometry described in the preceding section studies the polarity between finitaiy relations and finitary operations on a set A. Although each set of operations on A that is closed in the polarity is also a closed set of operations or clone on A, the converse need not hold. There are clones on infinite sets A that are not describable as the exact set of finitary operations on A preserving a given set of finitary relations on A. Example 1.5.1. Let (K, +, ■, 0,1) be an infinite field. Let T : ft -» N be the type of the polynomial operations on K. Thus ft is a clone on K. Let y be a non-diagonal, /-ary relation on K. It will be shown that y is not preserved by ft. Since (y, I) € DK, there is an m-ary operation f:Km -» K on K that does not preserve y. Thus there are elements (au,...,au) of y, for 1 <, i <, m, with (fljj... almf,..., an ... almf) € y. For 1 ^ /' ^ m, define the delta polynomial

S^,...,aiJTu...,Tm)=U

n

; - l ¡¡¡¡tice. {ahJ\l¿hzl)

[cf. (1.4.6), II (3.6)]. Note that ( * „ . . . , xm) e {(aA1 implies 5(fla a¡m) (xv..., xm) = if (x 1 ( ..., xm) = (an 0. Now consider the polynomial

a

ij

Z^r K

ahm) \ 1 <, h <. 1} aim) then 1 else

p(Tu...,Tm) L (*!

-0 e ««*!

(xx...xmf)8iXi

Xm)(

T

i>--->:rm)-

ühm)\l£h£l)

Note that p(,a¡¡,..., aim) = au ... aim f foi 1 £ i g I. Thus y is not preserved by the element p of ft. It follows that TS = DK and T SR = TK # T. In other words, the clone of polynomial operations on K is not closed in the polarity of Section 1.4 between finitary relations and finitary operations. D

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301

The main goal of this section, Theorem 1.5.2 below, is to show that each clone on a set A is realized as the set of operations preserving a single relation. However, two adjustments to the geometrical framework of the preceding section are needed. Firstly, consider an n-ary operation / : A" -* A on A. For an element i of {1,..., n), the i-th argument of / is said to be fictitious if V(alt . . . , «,_!, a l + i , . . . , a„) G A"~x, Va G A, Vft e A, ax...a¡_1aa¡+1...anf = a1...a¡_1baí+l...anf. One then identifies two operations / , g on A that can be obtained from one another by the addition or removal of fictitious arguments. For instance, if a0 is a fixed element of A, then one does not distinguish between the nullary operation A0 -» A; oo >-* aQ and the unary operation A -» A; a -» a0. The second adjustment is to admit "infinitary" relations, i.e. subsets of an infinite power of A. Note that if / is a possibly infinite index set, and A is an algebra of type T : fi -» N, then the power A1, realized as the set Set (/, A) of functions, inherits the componentwise operations (1.5.1)

) making it an algebra (A1, SI) of type r. An index set / or IA will be associated with each set A, such that there are mutually inverse bijections (1.5.2)

I -*AXl;x~*

(x\,xp)

A Xl^I;(a,x)

-*a°x.

and (1.5.3)

The inverse relationship between these bijections reduces to the following "identities": (a) (b) (c) !

VA: G / , x = x\°xp; V a e ^ , V i G / , ( f l o j t ) A = fl; Va<=A,Vx<El,(a°x)p = x.

Note that a non-empty set A is infinite iff there is a bijection A -* A X A. In this case, one takes the bijection to be (1.5.2), so that / = A for infinite A. If A is empty, one also takes / = A. Otherwise, A has a positive number d of elements, so there is a bijection ß : {0,1,..., d - 1} -* A. In this case, one takes / = N, and then (1.5.3) is given by a°x = aß'1 + dx. The Division Algorithm (cf. Section II 3.2) yields nkß~x as the remainder and np as the quotient on dividing the natural number n by d. Now for each natural number r, there is a surjection / -» Ar. If A is empty, it is the identity map on A. Otherwise, define the map (1.5.5)

/ -> Ar; x -» (*A, *pA

xpr~x\).

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CHAPTER IV UNIVERSAL ALGEBRA

For a in A, define L.(a)'- I -* I; x >-> a ° x. Fix an element xQ in /. Then (1.5.5) has the section (1.5.6)

sr : A' -> /; (a,

ar) -» al °(a 2 . ••• »( flr o * 0 ) . . . ).

In other words, the image of ( a „ . . . , ar) under (1.5.6) is xQ L,(ar)... L.(AJ). The fact that (1.5.6) is a section of (1.5.5) then follows by the identities (1.5.4). Extending (1.4.2) to the (potentially) infinite case, one says that a subset R of A1 is preserved by an operation / : A" -» A on A if /? is a subalgebra of the power (/Í7, {/}). Theorem 1.5.2 [I. Rosenberg, "A Classification of Universal Algebras by Infinitary Relations", Alg. Univ. 1 (1972), 350-353.] Let A be a set, with index set IA or I such that A X / s /. Let Cl be a clone of operations on A, of type T :Cl -* N. Then there is a subset R(A,ci) or R °f-^1 sucn t^uxt an operation fon A preserves R iff it belongs to the clone Cl. Proof. For empty A, take R = {1&). The result follows. Otherwise, set (1-5.7)

ÄM>n)-st(^,n),

the subalgebra of (A1,Cl) generated by {p r A|r e N). Note that R is preserved by the members of the clone fl. Conversely, an operation f: A" -* A that preserves R will be shown to lie in the clone fl. It is convenient to use prefix notation for operations. Since / preserves R, one obtains /(A, pk pn-1A) as an element of the subalgebra R of {A1, Cl) generated r by {p A | r e N). In other words, there is an element (k, pk,..., p"~lk) = <ú(av.... an). Thus / = 0. Vn e N, na = (n + p)a.

VARIETIES

303

(b) Suppose that each element of a subset R of A' is periodic. Show that an operation / : A" -» A on A preserves R iff it preserves a subset of (the graph of) the function LA of (1.4.1). 1.5C. (Continued Fractions.) Let A be the set of positive integers. Let / be the set of irrational numbers greater than 1. Show that the specifications a°x = a 4-x~ l ,

x\ = l*J,

*P= ( * - l * J ) - 1 (so that inverse Wright, Oxford,

x\ is the largest integer less than or equal to x) yield mutually bijections (1.5.2) and (1.5.3). [Cf. G. H. Hardy and E. M. An Introduction to the Theory of Numbers, Clarendon Press, 1979, Chapter X.]

2. VARIETIES A quasigroup Q may be construed as an algebra (ß, •, / , \ ) of signature <2,2,2>, i.e. equipped with three binary operations. Amongst all the algebras of such signature, quasigroups are distinguished by the requirement that the identities (I 2.2.3) have to hold. The class of all quasigroups is then seen to have special properties. For example, the direct product of two quasigroups is again a quasigroup, subalgebras of quasigroups are quasigroups, and homomorphic images of quasigroups are quasigroups. The universal algebraic concept of a "variety" (modeled somewhat on the concept of Exercise III 3.4L) abstracts these properties of the class of quasigroups. On the one hand, a variety will be the class of all algebras of a certain type that satisfy a certain set of identities. On the other hand, a variety will be closed under the formation of products, subalgebras, and homomorphic images. In parallel with consideration of varieties, one often has to take account of other classes of algebras. Recall that a quasigroup A was originally defined (Section I 2) as an algebra (Q, • ) of signature <2> in which all the right and left multiplications biject. Example I 2.2.1 showed that the class of all such algebras is not closed under the formation of homomorphic images. The study of varieties begins in Section 2.1 with the study of "prevarieties": classes required to be closed only under the formation of products and subalgebras. EXERCISES 2A. Construe quasigroups as algebras of signature <2,2,2> satisfying (12.2.3). Show that homomorphic images of quasigroups are quasigroups.

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CHAPTER TV UNIVERSAL ALGEBRA

2B. Construe quasigroups as algebras of signature (2) in which all the right and left multiplications biject. (a) Are products of quasigroups also quasigroups? (b) Are subalgebras of quasigroups also quasigroups? 2C. Construe unital rings as algebras (/?, - , 0 , •, 1) of type ({-,•} X {2}) U ({0,1} X{0}). (a) Are subalgebras of unital rings also rings? (b) Are subalgebras of fields also fields? 2.1. Replication and Prevarieties Throughout this section, let T : ÍI -* N be a finitary type. A class of r-algebras is a (non-empty) subclass Kp of the object class TJ, of the category r of T-algebras and r-homomorphisms. For such a subclass Z^, denote K_ to be the full subcategory of T whose object class is Kp. Thus K¡ is the class of all T-homomorphisms between algebras of the class Kp. Let G Ä or G : K-* T denote the inclusion functor from K to T. This is a "forgetful" functor". For example, its object part G : Kp -» T^, A -* A "forgets" that its argument A belonged to the subclass Kp. The main goal of the current section, Theorem 2.1.3 below, is to find conditions on Kp and K guaranteeing that there is an adjunction (R^G^, TJ, S) in which the components of the unit surject. Such an adjunction Is called a replication. The left adjoint R% or R is called the replica functor. The conditions to be imposed on the classes Kp involve the formation of subalgebras, products, homomorphic images, etc. It is convenient to set up a calculus for such constructions. Definition 2.1.1. Let Kp be a class of r-algebras. (h) A T-algebra is a member of the class HKp iff it is isomorphic to the image of a T-homomorphism whose domain is a member of Kp. (i) A T-algebra is a member of the class iKp iff it is isomorphic to a member of Kp. If Kp = ¡Kp, one also says that Kp is abstract. (p) A T-algebra is a member of the class PKp iff it is isomorphic to the product in T of a multiset of members of Kp. (Warning: Note carefully that the products are to be taken in T, and not in the category K. Cf. Exercise 2.1J.) (s) A T-algebra is a member of the class SKp iff it is isomorphic to a subalgebra of a member of Kp. The notation is extended to the corresponding full subcategories of T. For example, HK denotes the full subcategory with object class HKp. D Towards (p), it is helpful to recall the explicit construction used in Exercise O 3.2Q to realize the product n , e ¡A, of a multiset (A¡ | i e I) of sets. Taking

VARIETIES

305

p : LieIA¡ -* I as the disjoint union (coproduct in Set) of the maps A¡ -» {i}, the product is realized as the set (2.1.1)

L4\sp~h)

n^i-{*:'-»

of sections of p : L,ie ¡A{,-* I. One may then define a T-algebra structure 01,« ,Ai,a) on (2.1.1) by (2.1.2)

»'(*! ■ • ■ * « « )

" ( » l )

• • ■ ( » « ) «

for operators SI, sections s-, and indices i e /. The definition (2.1.2) ensures that the projections TTJ :YlíeIA¡ -*A¡; s *+js aie T-homomorphisms (Exercise 2.1A). Thus (fl¡e¡A¡,(l) is the product in T of the multiset < ( A,,, il ) | Í e / > of T-algebras. The operators of Definition 2.1.1 are usually written acting from the left. They obey the following commutation relations. Proposition 2.1.2. Let K^be a class of r-algebras. (o) Then PSK^ ç SPK^. (b) Then PHKp Q HPKp. (c) Then SHKp ç HSK^. Proof (a) Suppose V/ e /, B,<,At e Kp. Then UieIB¡ <. n i e / ^ , e PKp. (b) Suppose that for each i in /, the algebra B¡ is the image of a surjective homomorphism f¡ whose domain A¡ lies in Kp. Since f¡ : A¡ -» B¡ surjects, it has a section s¡ : B¡ -* A¡ (not necessarily a homomorphism) with s,f¡ = 1B . Consider the following diagram in Set:

riBiJii—>ru,

i el

i el

riK/;)

if£—»n*, IE/

(2.1.3) ■* A,

■* B,

For each i, the composite across the bottom row is 1 B . Thus the composite across the top row is n , e / l f l i = l n B i . In other words, the homomorphism nfe/iir./i) surjects, since it has n¡e¡(ir¡s¡) as a section in Set. Thus (c) By the First Isomorphism Theorem 1.2.7, it suffices to consider a subalgebra 5 of a quotient A" of an algebra A in K¡¡. Define B = (nat o) _ 1 (5). Then B is a subalgebra of A. Moreover, S = (uBa)a s ß<«n»2)

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CHAPTER IV UNIVERSAL ALGEBRA

by the Second Isomorphism Theorem 1.2.8(b). Thus S = B n a t ( a n ß 2 ) e f f S Kp, as required. D One may now examine necessary and sufficient conditions for the existence of replications. Theorem 2.13. Let T :Cl -* N beafinitary type. Let Kp be an abstract class of T-algebras, and let G : K-+ g be the inclusion in g of the full subcategory with object class Kp. Then the following conditions are equivalent: (a) £ , = SPKp; (b) Kis complete, G:K-+ gis continuous, and Kp = SKp-, (c) There is a replication (R, G, r¡, e). Proof, (a) =» (b). By Proposition 2.1.2(a), one has PKp = PSPKp £ SPPKp = SPKQ — Kp. Thus T-products of ¿^-algebras are jKp-products. By Theorem III 2.5.5, it then suffices to show that K has equalizers given by equalizers in h T. Consider a parallel pair AX^A2 of arrows in K. Then their set equalizer

~~

h

E = {a e Ax | afx = af2) is a subalgebra of Ax, and thus lies in SKp = SSPKp = SPKp, as required. (b) =» (c). Considering the first two conditions of (b), one observes that for the Freyd Adjoint Functor Theorem 3.5.2 to apply, it suffices to verify the solution set condition. Well, for a T-algebra A, the set (2.1.4)

(nat a : A - A" | Aa e K j

dominates the comma category (A, G). Indeed, for an object h : A -* BG of this category, the subalgebra Ah of B lies in SKp = Kp. The First Isomorphism Theorem 1.2.7 yields the commutative diagram A™L«.A.G

AhG

A —¡—>BG h in T. Since the upper arrow of the right hand side is an isomorphism, one has A" e Kp. Thus nat a belongs to the dominating set. Applying the Freyd Adjoint Functor Theorem, one obtains an adjunction (R, G, TJ, e). For a T-algebra A, consider the initial object r\A: A -* ARG of the comma category (A, G). Now Ai)A, as a subalgebra of the algebra AR in Kp, itself lies in

VARIETIES

307

Kp. Then T^ : A -* Ar^G is also an object of (A, G). Moreover, for an object h : A -* BG of (A, G), consider the unique (A, G)-morphism / or jG : (r)A :. A -* ARG) -*(h:A-* BG) given by the initiality of r^-.A-* ARG. One then obtains a unique (A, G)-morphism as the composite ArjA «-» AR —► B. Thus r^: A -* Ar\A is also an initial object of (A, G), so that the components of the unit in (J?, G, 17, e) may be taken as surjective. (c) => (a). Let A be a subalgebra of a member B of Kp, with inclusion j : A -* B. Consider the {A, G)-morphism given by the diagram VA

A — = - » ARG

fG . A

:—► BG

J

Since j — T ^ / G injects, it follows that the surjective map 7^ also injects, yielding an isomorphism of A with the member AR of Kg. Since Kg is abstract, A also lies in Kp. Thus SKp = Kp. Now let A be the product Y\ie,A¡ in T of members A¡ of Kg. Consider — the diagram A — ^ - ARG -£-+

A,

h Il

A = —

i,

A — - — * A¡

Here /, : (r)A : A -» ARG) -* (TT, : A -» A¡) is given by the initiality of r¡A in (A, G), while h : ARG -» A is given by ^4 = n ¡ 6 / j4 ( -. Then l^ir, = i)Af¡ = 7^/i7r( for each 1, whence T¿< A = 1A. Thus the surjective map r^ also injects, yielding an isomorphism of A with the member AR of Kg. As before, one — obtains A e Kp and thus PKp = Kp. D For a finitary type T : ft -» N, an abstract class Kp of r-algebras satisfying the equivalent conditions of Theorem 2.1.3, or the corresponding full subcategory K, is called a prevariety. For a r-algebra /I or (A, SI), the congruences a with A" in a prevariety Â^ are called K-congruences on ^4 [cf. (2.1.4)]. The subposet of CgiA, iîT induced on the set of ^-congruences is denoted by Cg^M, SI) or Cg s A. Note that the singleton r-algebra, as the product of an empty subset of"â prevariety Kp, lies in Kp. Thus A2 e Cg^A for each prevariety K. ~

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Proposition 2.1.4. Let Kbe a prevariety of r-algebras. Then for each r-algebra A, the poset Cg* A is a complete lattice, and a meet subsemilattice of CgA and } X {0} and the prevariety Kp of singleton subalgebras. Then for any set X, the free K-algebra XClRK, as a member of K^, is a singleton. Example 2.1.5. (a) (Convex Sets.) For the type r : I ' -» {2} of Example 1.3.4(c), one obtains the class Kp of convex sets as a prevariety. This is readily verified using condition (a) of Theorem 2.1.3. For a finite set X = {0,1,2,..., n}, the free ^-algebra over X is called an n-simplex. (Cf. Exercise 2.1D). (b) (Free Groups.) For the type (r : ft -» IM) = {(-,2),(-1,1),(1,0)}, consider the prevariety Gpn of groups. Then for a set X, the Gp-algebra XSÏ RQ. is isomorphic to the free group XG constructed in Section I 1.4. D A crucial property of free ^-algebras, their projectivity in K, is described by the following proposition (cf. Exercise II 2.3J). Proposition 2.1.6. Let Kbe a prevariety of r-algebras. Consider thefreeK-algebra XSIRK over a set X, and a surjective K-morphism g: A -» B. Then for each T-homomÔrphism h : XCIR,,-* B, there is at least one r-homomorphism Ji : XilRK-+ B such that the diagram X£IRK -J-+

A

(2.1.5) XilRK —:—► B ä h commutes. (One says that h lifts to h.)

VARIETIES

309

Proof. Consider the adjunction (ClRg, U, 77, e), and the diagram Vx

X

* XÍIRJJ hU

f

X

i

» AU gU

Vx

* > XSIRM ... » BU S hU in Set. Since gU surjects, there is an element / of SetíA", AU) making the bottom rectangle commute. The initiality of r\x in{X, U) then yields a T-homomorphism h : XSl R^-> A making the top rectangle commute. Using the bijection
EXERCISES 2.1A. Verify that (2.1.2) yields Y\ie,Ai as a product in T. 2.1B. Show that / acts as an identity for the operators of Definition 2.1.1. In other words, for each class Kp of T-algebras, verify ISKp = SK^, SlKp = S Kp, etc. 2.1C. Give an alternative proof of (b) =» (c) in Theorem 2.1.3 by showing ■qA = nat flCg^ A for a r-algebra A. 2.1D. Show that the (n - l)-simplex [in the sense of Example 2.1.5(a)] is isomorphic to the subalgebra of flR", / " ) generated by the subset {(1,0,0,...,0,0), (0,1,0,...,0,0),...,(0,0,0,...,0,1)}. 2.1E. Verify the claim of Example 2.1.5(b). 2.I.F. Let T : il -» M be the type {(-, 2), (0,0)}. Consider the following classes Kp of abelian groups G. For each, determine whether or not Kp is a prevariety. If Kp is a prevariety, describe 77 : A -» ARK for an abelian group A. °° (a) Kp = {G\ Tor G = {0}}, the class of torsion-free abelian groups (cf. Exercise II 3.3.D). (b) Kp = {G I Tor G = G), the class of torsion abelian groups. (c) Fix an integer n. Then Kp = {G \ Gn = {0}}. (d) Kp = {G\3n^Z. Gn = {0)1 2.1G. For each prevariety Kp in Exercise 2.1F, describe the free ^-algebra on a set X.

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CHAPTER rv UNIVERSAL ALGEBRA

2.1H. Prove that Kp is a prevariety iff Kp = SKp and Kp = PKp. 2.11. Let (r:ft-*N) = {(•, 2)}. Let Kp denote the class of commutative semigroups. (a) Show that Kp is a prevariety. (b) Describe the free ^-algebra XCIR^ on a set X. (c) Show that XSIR* is isomorphic with the semigroup X*K - {1} of Exercise I 1.5E. 2.1J. Let A be a principal ideal domain. Let (T : il -* N) be the type {(-,2)} U G4 X {1}) U {(0,0)}. Let Kp be the class of torsion modules : /1-modules W with W = Tor W (cf. Exercise II 3.3D). Show that K is not a prevariety in T, although £ is complete and Kp = SKQ. 2.1K. Set n = 6 in Exercise 2.1F(c). Show that Z 3 is projective in K, but not free in K. (Hint: The order of a finite free X-algebra is a power of 6.) 2.1L. Let K be a unital, commutative ring. (a) Construe the category of jK-algebras as a prevariety Alg,,. (b) Show that the polynomial algebra K[T] is the free Alg^-algebra over the singleton {T). [Hint: (II 3.4).] (c) Is K[TV T2] the free Algr-algebra over the doubleton {7\, T2)l 2.2. Bicompleteness Throughout this section, let T : fí -» (^J be a finitary type. Theorem 2.1.3 showed that any prevariety K of r-algebras, including T itself, is (small-) complete. The main result of this section, Theorem 2.2.3 below, shows that K — is also (small-) cocomplete. The key task is to show that T is cocomplete. By Corollary III 2.5.6, it suffices to exhibit coequalizers and coproducts in T. The construction of coequalizers is immediate. Given a parallel pair (/, g) of T-homomorphisms A -* B, let K be the congruence on B generated by the binary relation {(af,ag) \a e/1}. Then the coequalizer of (/,g) is the natural projection nat K: Az^B^^B«.

(2.2.1)

g

Indeed, if fh = gh for a r-morphism h : B -* C, the kernel ker h lies in the interval [K, B2] of CgB. By the Third Isomorphism Theorem 1.2.9, there is a unique congruence ß on B" with (nat K ) * ( / 3 ) = ker A. Consider the commutative diagram B" « " a t " (2.2.2)

nat ß

B

nat ker h

*J

k hh (BKx\ß )P—:—►- BDker " —:—» Bh

VARIETIES

311

in T, where the right-hand square is an instance of (1.2.5), and /, is the isomorphism of (1.2.8). Then (nat *X(nat ß)ixi2j] = h, verifying that (2.2.1) is a coequalizer in r. Example 2.2.1 (Free Presentations). Let A be a r-algebra lying in a prevariety K. Let eA : A0 -* A be the component at A of the counit e in the adjunction (ÍIR& U, r¡, e). Thus A0 = AUilRr is the free JÇ-algebra over the underlying set AU of A. Set Ax = ker eA. Thus Ax = {(a, b) &A\\ aeA = beA). Note Ax e SPKQ = /Çg, using condition (a) of Theorem 2.1.3. Define d0 : Ax -> A0; (a, b) -* a and dx : Ax -* A0; (a, b) -* b. One obtains a parallel pair (d0, dx) of /C-morphisms Ax -* A0. Then the congruence K of the coequalizer construction in T is (the congruence on A0 generated by) the binary relation {(axd0, axdx) \ o, e Ax] = {(a, b) e. A\\ (a, 6) e kerfi^}= ker eA. Thus (2.2.3)

^ ^ ^ o - ^ ^ l

is a coequalizer diagram, both in T and JL. It is called the free presentation of the K-algebra A. As an instance of the uniqueness of coequalizers to within isomorphism, the algebra A is specified by the parallel pair (d0, dx) to within isomorphism. The elements (a, b) of Ax, sometimes written in the form "a = b", are called the relations of A (relative to /p. D Now let F : / -* T¿ be a function. For each element j of /, consider the free presentation jFx =5yF0 -* jF of the r-algebra jF. Let e¡ : jFU -» 5 be the insertion of the underlying set jFU of jF into the disjoint union S = VjejjFU. Apply the functor fi to get the r-homomorphism e ; il : jF0 -* Sil. Define K to be the congruence on Sil generated by (2.2.4)

(J

ye/

{{axdüef,axdxe?)\axejFx).

Now for each j in /, one has d 0 e"nat K = dxefviai K. Since the free presentation of jF is the coequalizer of the pair (d0,dx), one obtains a unique r-homomorphism i¡ : jF -» SílK with e¡Fi¡ = ef nat K. Then the rhomomorphism i; : jF -» 5ÍÍ" is the insertion of /F into the coproduct Sil" = LjejjF of the yF in r. Informally, the construction of LjeJjF takes the free r-algebra 5ÍÍ on the disjoint union S of the underlying sets jFU of the jF, and then "imposes the relations of each jF relative to r." With coequalizers and coproducts now constructed in r, Corollary III 2.5.6 yields the following: Proposition 2.2.2. The category r is cocomplete. D The main cocompleteness result is then immediate.

312

CHAPTER rv

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Theorem 2.2.3. Let T : il -» M be afinitarytype, and let K be a prevariety in T. Then the category K is cocomplete. Proof. Let F : J -» K be a functor with small domain. Since T is cocomplete, the composite FG of F:J->K with the inclusion G :K-* T has a colimit iFG : FG -» (lim FG)A. Consider the replication (R, G, TJ, e). Define U15 F = (liig FG)R. Define a natural transformation t,F : F - » ( Hig F)A with component if = (lin) F)A is the colimit of the functor F : / -» #. D EXERCISES 2.2A. Determine the free presentation of the abelian group Z 2 in the prevariety of abelian groups A with 4A = 0. 2.2B. Let F : / -» ¿j be a function. Let A be a r-algebra. Given r-homomorphisms /y :jF-*A for each / in /, show how to construct the unique T-homomorphism E ; e y/7 : E ; e y)F->/I with t,Ey e y/ ; =/) for each 1 in J. 2.2C. Let X and Y be sets, and let K be a prevariety. Prove that the free ^-algebra over X U Y is the coproduct of the free ^-algebras over X andy. 2.2D. In the proof of Theorem 2.2.3, verify that J : F -» ( liig F)A is indeed an initial object of F / Kb. 2.2E. Let T : fi -» M be a finitary type, and let (/l, il) be a r-algebra. (a) Show that SP{A) is a prevariety. (b) Show that SP{A} is contained in each prevariety Kf¡ of which A is a member. (Thus SPM) is called the prevariety generated by A). (c) The logarithmetic S?A of y4 is the subalgebra of AAU generated by 1AU : AU -* AU. Prove that A 4 is the free SPMJ-algebra over the singleton {lAU). (d) Show that the free SPM}-algebra over a set X is the coproduct Z,X2A. 2.2F. Determine the logarithme tics of the groups Z 4 and S3. 2.2G. Determine the logarithmetic of the quasigroup (Q, °) of Exercise I 2.1F. 2.2H. (Tensor Products of tf-algebras.) Let K be a unital commutative ring. Let T = (K X {1}) U ({-,m} X {2}) U ({0,1} X {0}) be the type of ^-algebras. (a) Interpret the category CAlgy of commutative .K-algebras as a prevariety of r-algebras. (b) For ^-algebras A and B, show that the tensor product AK ®KBK of the .K-modules AK and BK becomes a .K-algebra A ® B under the multiplication given on primitive elements by axa2 ® b1b2 = («j ® f>iXc2 ® f>2).

VARIETIES

313

(c) If A and B are commutative, show that A ® B with the insertions A -* / 4 ® 2 ? ; a - » a ® l and B -* A ® B; b *-» 1 ® ¿> is the coproduct of ^4 and 5 in the prevariety CAlgy. (d) Prove K[Tt, T2] s KITJ ® K[T2] (cf. Exercise 2.1L). 2.3. Satisfaction and Varieties Consider the type T = {ß, 8, y) x {2} of signature <2,2,2> used for quasigroups in Example 1.3.4(b). Amongst all the algebras of type T, quasigroups are distinguished by satisfaction of the identities (I 2.2.3). For a quasigroup (Q, ß, 8, y), (I 2.2.3) (IR) takes the form (2.3.1)

Xlx2ßx28

= xy

In the terminology of Section 1.3, the left-hand side of (2.3.1) is the derived operator u = xxx2 ßx28. The right-hand side is the derived operator v = xx. Satisfaction of (I 2.2.3) (//?) or (2.3.1) by a quasigroup (Q, ß,y,8) means precisely that the binary clone operations (M, 2) ß or uQ : Q2 -» Q; (p, q) <-* pqßqS and (v, 2)Q or vQ : Q2 -» Q; (p, q) <-> p coincide. For the rest of this section, fix a finitary type r : il -» N. An identity (u, v) or u = v of type T is a pair («, y) of derived operators, i.e. an element of the direct square PCI2 of the word algebra PCI on the set P (1.3.4) of positive integers. A r-algebra A is said to satisfy the identity u = v if the n-ary clone operations (u, n)A and (u, n)A on ^4 coincide, where n = max ((arg w) U (arg v)). One also says that the words u and v are synonymous in A For example, the magma words xxx2 fix3 ¿i and xlx2x3 fi.fi are synonymous in each semigroup. Proposition 2.3.1. Leí A be a r-algebra, and let u = v be an identity. The algebra A satisfies u = v iff V / e r (PCI, A), uf = vf. Proof. Set n = max ((arg u) U (arg u)\ and X = {*„..., xn). If ^4 satisfies u = v and / e riPfi, A), then uf = x{...xfn(u,n)A =x{...xfn(v,n)A = uf. Conversely, suppose V / G T ( P Í Í , A), uf = vf. If y4 is empty, then Cl contains no nuUary operations. By Proposition 1.3.3, there are then no nullary derived operations. Thus n > 0, and (u, n)A = lg, = (v, n)A. Otherwise, let aQ be an element of A. Given a e Set (A", A), define 5 = (a : X -* ^ ) U«/» - * ) -» {a0}). For / = 5 % e T ( P Î Î , ^ ) , one then has x\ ... xan(u, n)A = uaaeA = uf= vf= x\ ... xan(v, n)A. D Satisfaction of identities is preserved by the constructions of Definition 2.1.1: Proposition 2.3.2. Let Kp be the class of r-algebras satisfying an identity u = v. Then: (h) HK^^Kp; (p) PKp = Kp; (s) SKp^Kp.

314

CHAPTER IV UNIVERSAL ALGEBRA

Proof, (h) Suppose A satisfies u = u, and that g: A -* B is a surjective T-homomorphism. By Proposition 2.1.6 applied to the improper prevariety r, the free algebra PCI is projective in r. Thus each element h of T(PCI, B) lifts to Ji in ¡(PCI, A) with h = Äg. Then uh = wÄg = vhg = u/t, SO that B satisfies u = v. (p) Suppose each element A¡ of a multiset {A, \ i e /> of T-algebras satisfies u = v. Then for / e T(PCI, FI; 6 /^4y), one has u/7r, = vfir¡ for each i in /, whence uf = vf and YljeIAj satisfies u = v. (s) Suppose that A satisfies u = v. Let j:B-*A be injective. Then for f & T(PCI,B\ the equality ufj = vfj implies uf=vf, so that 5 satisfies u =~v. D Proposition 2.3.2 shows that the class of r-algebras satisfying a given identity is a prevariety. More generally, the class Kp of r-algebras satisfying a given set of identities is a prevariety. In addition, Kp = HKp. A variety is defined to be a prevariety KoxKp satisfying this additional hypothesis Kp = HKp. By Proposition 2.1.2, Kp is a variety iff J£„ = HSPKp (Exercise 2.3ÁJ. The crucial property of varieties is given by a classical converse to Proposition 2.3.2: BirkhofTs Variety Theorem 2.33. Let r : il -» N be afinitarytype. Let Kp be a class of r-algebras. Then Kp is a variety iff it is the class of T-algebras satisfying a certain set of identities. Proof The " i f direction follows by Proposition 2.3.2. Conversely, suppose that J£o is a variety. As a prevariety, K possesses a replication (R^, Gg, 17^, e&). It will be shown that Kp is the class of r-algebras satisfying the set kerÍT/^n : Píl -» PÍIRG) of identities. Suppose first that A is a member of Kp. Each r-homomorphism / : PCI -» AG factorizes as / = y-puif*^)0, so that uf = vf for each identity (u, v) in kerrç^n. Conversely, suppose that A is a r-algebra satisfying ker i7=PiiConsider the adjunction (il, U,rç,e). It will be shown that ^&x{t]=AUCl : AUtl -* AUSIRG) <. ker (e,, : AUCl -» A). Then A, as a homomorphic image of the ^-algebra AUÜRG, will itself be seen to lie in Kp. For an element (wvw2) of ker r)-Aun, let (arg Wj) U (arg w2) = {a!,..., a„) = B with |B| = n. Define j:B-+P; a¡ -> oc,-. By the naturality of the unit 17 in the replication (R, G, 17, e) at the r-homomorphism y'O, one has the commuting diagram BÎÏ

-^ BCIRG jilRG

;fl ■

1

<

f

il —-—► PÍIRG

VARIETIES

315

in T. Since (wu w2) e kerTj- Bn , one has (u,v) == (wja,w2ja) <= kerrj^n. Define a:{xr xn)-* A; x¡-* a¡. Since A satisfies u = v, one obtains wleA = xax ... x"n(u, n)A =x"x... xan{v, n)A = w2eA. Thus (w„ w2) e ker eA. D The proof of Birkhoff s Variety Theorem 2.3.3 shows that each variety K of T-algebras, with replication (R^, Gg, i A e&), is specified uniquely as the class of T-algebras satisfying ths~equational theory Eq(Ap, the relation (2.3.2)

Eq (K) = ker (r^Pil : Pil -

PSIR^J^

of synonymity in the free ^-algebra over P. For a variety L contained in the variety K, the Third Isomorphism Theorem 1.2.9 yields a congruence Eq (L; i p on the free algebra PÜRK in K with (nat E q ® ) * Eq(L;|p = Eq(£J, as displayed in the following diagram: *-nat Eq(£)

(2.3.3)

r-nat Eq(L;£)

pn -——-* P Í Í ^ /-nat Eq(¿)

k

Note that Eq (L; T) = Eq (L). A natural question arises: Which congruences on PCIRK are of the form Eq(L; Jp for a subvariety L of a variety JRT? TO answer thus question, consider a T-algebra (A, ft). The monoid T(A,A) of endomorphisms of (A, il) may be construed as a domain of unary operations on A. Then congruences of the algebra (A, fi U T(A, A)) aie called fully invariant congruences of the algebra (A, fi). Theorem 2.3.4. Let r : fi -» N fee A finitary type. Let K be a variety of T-algebras. Then a congruence on PilR^ is fully invariant iff it is of the form Eq (L; #p for a subvariety L^of K. Proof. Use the notation of (2.3.3). First, consider an identity (u, v) with (uk, vk) e Eq (|C;L), and an endomorphism t of PSIRg. By the initiality of I. PCi -* PilRLGL in (Pfi,G L ), there is an endomorphism t' of PilRL with far = It'. Since (w/, U/) e Eq(L), one has wtor = ult' = u/í7 = ufar. Thus (ukt, vkt) e Eq (|C; L), so that Eq (AT; L) is fully invariant. Conversely, let 6 be a fully invariant congruence of PCIRK. Let L be the variety of all T-algebras satisfying k*9. It will be shown that k*8= EqL, whence 8 = Eq(L;|p. Certainly &*0 ^ EqL. Conversely, consider the nonempty ^-algebra A = PClRKe. It will be shown that A lies in L, and that A satisfies no identity in Pfi 2 - k*9. Thus EqL< k*6.

316

CHAPTER rv UNIVERSAL ALGEBRA

Consider an identity (u, v) with (uk, vk) G 0, and an element / of A) or of (PCl,GK)Q. By the initiality of k in (PCl,GK), there is a K-morphism /i such that The diagram T(PCI,

PCI

k

» PÙRgG£ hGK V

/>a — 7 — » AGK commutes. By the projectivity of PCIR^ in K (Proposition 2.1.6), there is then an endomorphism h of PCIR^ such"that the diagram PCM« —L_» PCiR^ nat 6 PCIRK

—► A

commutes. The full invariance of 9 then yields uf = ukh = ukhinat 6) = vf, so that A satisfies u = v. Thus A G LQ. Now consider («, u) G PCI2 - k*0, i.e. (ufc, vk) € 6. For *(nat 6):PCl-* A, one has M&(nat 6) ¥= ufc(nat 0), so that u = v is not satisfied by the L-algebra A. D Corollary 2.3.5 (Completeness Theorem of Equational Logic). A relation 2 on PCI is an equational theory iff it is closed under the following deduction rules: (a) V i > 0 , ( * , , * , ) £ £ ; (b) (u, v) G 2 => (v, u) G 2; (c) [(u, v) G 2 and (u, w) G 2] =» (M, W) G 2; (d) Vto G n , [VI <. i <í wT, («,-, v¡) G 2] => ( u t . . . uWT w, Ü, ... v„ u(are »>, (w,... ^„(M, n^n, w,... wn(ti, n) p n ) G 2. /Voo/ The relation 2 is closed under the deduction rules iff it is a fully invariant congruence of PCI. For example, closure under (e) corresponds to closure under an endomorphism / of PCI with xj = w¡ for 1 ^ i' ^ n, while (c) corresponds to transitivity. D

VARIETIES

317

EXERCISES 2.3A. Show that Kp is a variety iff Kp = HSPKp. 2.3B. Let Kp be a class of r-algebras. (a) Show that HSPKp is a variety. (b) If a variety L^, contains Kp, show that it also contains HSPKp. 2.3C. Let T be the type {(e,0)} U ({/3, y, 5} X {2}) of loops. Consider the derivation of the left inverse property x, = x2eyx2xi ßß from the First Moufang Identity xtx2 ßxt ßx3 ß = xix2xix3 ßßß, from (I 2.2.3) IL in the form x1x1x2 ßy = x2, and from (I 2.2.3) SL in the form xlx1x2yß = x2, as given in the proof of Proposition I 4.1.5. Write out the derivation as a step-by-step application of the deduction rules of Corollary 2.3.5. How many steps are required? 2J D . Let I denote the closed real unit interval [0,1] and I * the open unit interval (0,1). Set T = 1° x {2}, the type of Example 1.3.4(c). Let Kp be the class of convex sets, as in Example 2.1.5(a). (a) Show that finite /^-algebras have at most one element. (b) Show that 0 = (I o Xl')u{(0,0),(1,1)} is a fully invariant congruence on (I, I ' ). (c) Show that I e is not a member of the class Kp. 2 J E . (Axiom Rank and Base Rank.) Let K be a variety of algebras of type T : fi -* N. For each positive integer n, define Kn=HSP{{xl,...,x„}ílR¿ and K" = HSP[Pñnat{E<\{K) n {xlt...,

xn)il2)}.

(a) Prove £, ç K2 ç ••• ç £ ç • •• ç K2 ç K1. (b) Prove K= Un>0Kn= nn>0Kn. The axiom rank of K is ra(K) = min{n\K = Kn}. The base rank of K is rb(K) = min{n\K = Kn}. (c) Assuming the fact that the free group on 2 generators contains a subgroup that is free on a countable set, prove that the base rank of the variety of groups is 2.

318

CHAPTER IV UNIVERSAL ALGEBRA

(d) Assuming the fact that 2-generated subloops of the loops Q of Examples I 4.1.3 and I 4.1.4 are associative, prove that the axiom rank of the variety of groups is 3. 2.3F. Let T : il -> N be a finitary type. Let A be a T-algebra generated by a subset X. Show that A is the free algebra over X in a variety K of T-algebras iff each function / : X -* A is the restriction to X of an endomorphism / of (A, il). 2JG. Determine the two varieties of algebras of empty type. 2.3H. Let K be a variety containing an algebra of cardinality greater than 1. Show that K contains a simple algebra of cardinality greater than 1. 2.31. A Steiner triple system (S, B) is a set S together with a set B of 3-element subsets of S, such that each 2-element subset of S is a subset of a unique element of B. The elements of B are called triples or blocks. (Thus the crews of Example I 2.5 are the blocks of a 7-element Steiner triple system.) (a) Let STS be the variety of magmas satisfying the identities xxxli*.= xl,xlx2ix=x1xlix and xlx2fix2fj.= x2. For each element S of STS». show that there is a Steiner triple system (5, B) with B

- {{x,y,xyv)\x*y

eS).

(b) If (5, B) is a Steiner triple system, show that a magma (S, fi) is defined by xxfi = x and {x, y, xyfi) e B for x ¥= y. Furthermore, show that (S, /¿) is an object of STS. (c) Let M be the submonoid of (N, •, 1) generated by {3,7}. Show that n e M ^ > B S e S T S ^ l S l = n. [Hint: Use (I 2.4) together with //STS= S STj¡= fSTS= STS.] 2.4. Entropie Algebras and Tensor Products Fix a finitary type T : il -* N. Consider an algebra (A, il) of type T. For each operator w from il, the corresponding operation (2.4.1)

(o : A"" -> A

is a map to A from the direct power Aur of the set A. Generalizing from Exercise I 2.2M, the algebra A is said to be entropie if each operation w gives a r-homomorphism (2.4.2)

ü>:(A,il)'°r

- {A, il)

to {A, il) from the direct power (A, il)"" of the algebra (A, il). Thus by

VARIETIES

319

(1.2.1), the algebra (A, fi) is entropie iff it satisfies the identity (2.4.3)

( x , . . . xUT a>) ... (*(„T-i)uT+i - -. *„T.WT w) = (x1... x^T_i)uT+1(A) •.. (xUT... x„,T.aTil/)w

for each subset {i^, a>} of ÎÎ. In particular, Proposition 2.3.2 yields part (a) of the following: Proposition 2.4.1. (a) Let K^ be a class of entropie r-algebras. Then each member of HSPKQ is entropie. (b) Reducís of entropie algebras are entropie. D Example 2.4.2. (a) (Modules over Commutative, Unital Rings.) Let 5 be a unital ring. Consider the type {(0,0), ( - , 2)} U (S X {1}) of (right) 5-modules. For a subset {s, t] of 5, (2.4.3) reduces to the identity xxts =xtst. Applying the unary derived operations on each side of the identity to 1 in S, one sees that the right 5-module Ss is entropie only if S is commutative. Conversely, if 5 is commutative, then each 5-module is entropie. (b) (Convex Sets.) By (a), real vector spaces are entropie. By Proposition 2.4.1(b), reducts of real vector spaces are entropie. Then by Proposition 2.4.1(a), convex sets [as in Example 1.3.4(c)] are entropie. D Abelian groups are entropie [take S = Z in Example 2.4.2(a)]. In Example III 2.4.3, abelian groups were identified as groups in the category of groups. One may abstract this observation from groups to universal algebra. Definition 2.4.3. Let C be a category with finite products. Then a r-algebra in the category C is an object A of C, equipped with a C-morphism a : AaT -* A for each to in il. D Using Definition 2.4.3, entropie r-algebras are characterized as T-algebras in the category T. Consider a r-algebra A in a category C with finite products. The basic operations CD of the type r act on A as C-morphisms w : y4UT -» A. Concatenating the actions of basic operations, one then obtains A as a r-algebra in the category C. [For example, diagonals in (III 2.4.3) provide derived unary operations xxxxJ^ and xiJx1 fi.] The r-algebra A in C is said to satisfy an identity u = v if the /i-ary clone operations (u, n) : A" -» A and (v, n) : A" -» A are equal elements of C(A", A), for n = max((arg u) U (arg i>)). Using the Birkhoff Variety Theorem 2.3.3, one may then refine Definition 2.4.3. Definition 2.4.4. Let C be a category with finite products. Let K be a variety of r-algebras. (a) A K-algebra in the category C is a r-algebra A in C satisfying each member of the equational theory Eq (X) of identities characterizing K.

320

CHAPTER IV UNIVERSAL ALGEBRA

(b) A K-algebra morphism in the category C is a C-morphism / : A -* B between tween K-a\geb ^-algebras in the category C such that, for each operation w, the diagram »A (2-4.4)

f

(/,..-,/)

BaT — — * B in C commutes. (c) The category C ® K of K-algebras in the category C is the subcategory of C consisting of ^-algebras and K-algebra morphisms in the category C. D Example 2.4.5. For a group G and unital ring 5, consider an ¿-representation R:G -* Mod 5 of G, with corresponding module 1R = M. Then the split extension G[RM is an 5-module in the slice category Gp/G. The unary operations corresponding to elements J of J are morphisms s : (g, m) -* (g, ms). D For a r-algebra A and a set X, the set Set (X, A) of maps from X to A carries the T-algebra (A, (l)x. One may then recognize entropicity of A by the closure of sets of r-homomorphisms into A under the operations of Proposition 2.4.6. A T-algebra A is entropie iff, for each T-algebra X, the set T(X, A) is a subalgebra of Ax. Proof. Let i// be an m-ary operation and n. One obtains («l/l-"/»---(

a

m/l

••-/»
= (a,/, ... o,/„a) ... (amf1... amfn) a> = =

(a1...amil>f1)...(al...am (a1-..amtl,)...(a1...am^)(fi...fnw),

so that fx... fn(o is again in T(X, A), as required. Conversely, suppose that T(X, A) is a subalgebra of Ax for each r-algebra X. Consider the projections tri : (a,,..., a„) -» a, in T M " , A). Then for

VARIETIES

(ai

amn)^Amn,

321

one has

( a , . . . an(o) ... (ß ( m _ 1 ) n + , ...amn(o)ilf = (al,...,an)(irl = =

.••-n- n «').--(a( m -i) n+ i

a mn )(7r, ...-7rnw)t/r

(fl,,...,an)...(a (ûl--û(m-l)«

+

l'/'.••■.

a

n

-

û

m

n

'/')(

i r

l

•••■»■»<»)

= (fll---a(m-l)n + l,/')---(íln--am«
so that y4 satisfies the identities (2.4.3). D For the rest of this section, let K be a prevariety of entropie r-algebras. Proposition 2.4.6 shows that for each fixed K-algebra Y, there is a functor Ry : K-* K with morphism part ( / : X -> X'J" (K(Y, X) -> K(Y, X'); g ~ gf). The functor RY is a generalization of the functor RY of Section III 3.6 defined on the category of modules over a commutative, unital ring K, since this category is itself a prevariety of entropie algebras. The functor RY of Section III 3.6 was shown to have a left adjoint SY, whose object part assigned the tensor product Z ®KY to each ¿f-module Z. Now the general functor RY:K-*K also possesses a left adjoint SY :K-* K. The proof carries over almost verbatim from Section III 3.6, replacing K and Mod^ by K as appropriate. The set dominating the comma category (Z, RY) of r-bihomomorphisms Z X Y -* A- is (2.4.5)

U

K(Z,K{Y,Ma)),

oeCg^M

the union being taken over the set of ^-congruences a on the free ^-algebra M over the set Z X Y. One replaces quotients M/N in Section HI 3.6 with quotients M" by such congruences, for example in (III 3.6.2) and (III 3.6.3). Summarizing: Theorem 2.4.7. Let K be a prevariety of entropie r-algebras. Let Y be a K-algebra. Then the functor RY:K-*K with morphism part ( / : X -* X') -* (K(Y, X) -► K(Y, X')\ g^gf) possesses a left adjoint SY : K-* K. D The image ZSY of a AT-algebra Z under the functor SY is called the tensor product algebra Z ® ¿Y or Z ® Y. Some key properties of the tensor product are listed in the following: Proposition 2.4.8. Let K be a prevariety of entropie r-algebras. (a) For K-algebras X, Y and Z, the commutative law X ®Y =Y ® X and associative law (X ® Y) ® Z s X ® (Y ® Z) hold. (b) For a K-algebra Y and a multiset (Z¡ \ i G /> of K-algebras, the distributive lawY® LI, e /Z, s LI, e , y ® Z, holds.

322

CHAPTER IV UNIVERSAL ALGEBRA

(c) For a K-algebra Y and the free ¡^algebra TÙRK over a singleton set T (terminal object of Set), there is a natural isomorphism Y ® TílR^s Y. (d) Tensor products of free K-algebras are free. Indeed (f~X J)QRK& ICIRK ®JCIRK for sets I and J. Proof, (b) follows by cocontinuity of the left adjoint SY [Theorem HI 3.4.1(b)]. Note that K is cocomplete, by Theorem 2.2.3. The other results follow by the uniqueness of adjoints to within isomorphism. Note that (a) is the analogue of (III 3.6.6) and Exercise HI 3.6B, that (c) is the analogue of Example III 3.6.2, and that (d) is the analogue of Example III 3.6.3. D An interesting application of the concept of tensor product of entropie algebras is obtained by taking K to be the prevariety of convex sets, as in Example 2.1.5(a). If a finite set X represents the set of (pure) states of a (deterministic) system, then the free K-algebra XClRg over X [the (\X\ - 1)simplex in the terminology of Example 2.1.5(a) and Exercise 2.1D] represents the set of probabilistically mixed states of a corresponding non-deterministic system. For instance, given pure states x and y, and for p e 1° = (0,1), the state xyp [notation of Example 1.3.4(c)] is the mixed state yielding y with probability p and yielding x with probability 1 - p. Now if A and B are convex sets representing probabilistically mixed systems in this sense, the direct product A X B represents a combined system in which the constituents A and B remain independent. On the other hand, the tensor product A ® B represents a combined system in which the constituents A and B are coupled, no longer remaining independent. As a simple example, take A = {aua2}SlRK and B = {bit b2)ilR^. Compare the mixed state (O.OlXa,, &,) + (0.97XÛ!, b2) + (0.0lXa2, *>x) + (0.0lXfl2. ft2> of A <$ B with the mixed state (0.98ai + 0.02a2,0.506, + 0.50fc2) of A X B. Note that one is most likely to observe al in the constituent A of A ® B, and then one is most likely to observe b2 in its second constituent. On the other hand, although one is most likely to observe ax in the first constituent of A x B, this does not make any one state of the second constituent more likely than the other. The distributions of the states in the constituents are coupled in A ® B but independent in A x B.

EXERCISES 2.4A. Prove part (b) of Proposition 2.4.1. 2.4B. Give a complete verification that unital modules over a commutative ring are entropie. 2.4C. Give a direct verification that convex sets (A,l°) are entropie [cf. Example 1.3.4(c)]. 2.4D. Let M be a monoid. Show that each M-set is entropie iff M is commutative.

ALGEBRAIC THEORIES

323

2.4E. Verify the details of Example 2.4.5. 2.4F. Write out a proof of Theorem 2.4.7. 2.4G. (a) Show that commutative monoids are entropie. (b) In Section 11.5, the free commutative monoid X*K on a set X was interpreted as the set of finite multisubsets of X. Given sets X and Y, interpret the coproduct X*K ffi Y*K and the tensor product X*K® Y*\ 2.4H. (a) Show that semilattices are entropie magmas. (b) Show that the poset of non-empty subsets of a finite set X is the free (join) semilattice over X. (c) Let S be the join semilattice with Hasse diagram a —» b «— c. Show that S ® S has 15 elements. 2.41. Let X be a finite set, and let K be the prevariety of convex sets. If XÍIRK represents the mixed states of a system with state space X, show that the pure states are represented by the minimal non-empty walls of the algebra (X(IRK, I ' ). 2.4J. For T-algebras A, B, C in a prevariety K of entropie algebras, determine whether or not the "mixed associative law" (A X B) ® C = A X (B ® C) is satisfied. 2.4L. Let C be a locally small category with finite products. Let K be a variety of r-algebras. Show that a C-object A is a ^-algebra in C iff, for each C-object Y, the set C(Y, A) lies in K. [Hint: C(Y, AYr = C(Y, A»r).] 2.4M. Let M be a monoid. Show that an M-set in Setop is an A/op-set in Set. 2.4N. An undirected graph T is a tree if each pair of vertices in T is connected by a unique path of edges followed without retracing. The centroid of a triple (x, y, z) of vertices in a tree T is the unique vertex common to the paths connecting x with y, y with z, and z with x. (a) Show that (T, y) is an algebra of type {(y, 3)} under the operation (x, y, z) -» (JE, y, z)y mapping a triple of vertices to its centroid. (b) Show that there is a tree T for which (7\ y) is not entropie. 3. ALGEBRAIC THEORIES Let 5 be a unital ring. Let P = {x¡ < x2 < •••} be the set of positive integers, written as in (1.3.4). For each natural number n, consider the free (right) 5-module nS on the initial subset {xv..., xn} of P. Elements of nS have the form (3.1)

x1sl +x2s2 + ••• +x,,sn

with s¡ e 5. In other words, nS is the set of n-linear combinations (cf.

324

CHAPTER IV UNIVERSAL ALGEBRA

Example II 2.3.1). Note that 05 is the singleton {0}. Now let 5 (same symbol as for the ring) denote the full subcategory of Mode whose set 5 0 of objects is [nS|«eM}. This category is called the algebraic theory 5. It is small. Moreover, each object nS of 5 is the coproduct of the finite number n of copies of 15. For positive integers / and m, the morphism set S(IS, mS) may be identified with the set 5,m of (/ X m)-matrices over the ring 5. Thus the algebraic theory 5 comprises both the set Sx of matrices over the ring 5 and the set 5 0 of indexed sets of linear combinations over the ring 5. Now consider a right 5-module V. For each natural number m, recall the isomorphism Vm = 5(5, V)m s S(mS, V) of Section II 2.3 and Exercise II 2H. Under this isomorphism, the element (vx,...,vm) of Vm is identified with the unique 5-module homomorphism v : mS -* V; xxsx + ••• +xmsm -* vxsx + ■•• +vmsm specified by the substitutions x¡ -» v¡ for 1 á i: ^ m. Each morphism / : /S -> mS in the algebraic theory 5 yields a map (3.2)

S(f,V):S(mS,V) -+S(lS,V);v ~ fv.

In particular, the morphism A: 15 -> mS;xx <-*xxsx + ■■• +xmsm yields the map S(A, V) : S(mS, V) -> 5(15, V); v -» Xu that takes the form (3.3)

S(\,V):Vm

- V1;(u1,...,um)

- uxsx + - +umsm

of the corresponding m-ary linear combination on identifying v with ( u „ . . . , 0 and Air. I S - » K ; JC, -» x^sx + ■■■ +xmsm -* «,*, + ••• +vmsm with v1sl + ■■■ +vmsm [cf. (II 2.3.1) with / = {1,..., m)\ As in (II 2.1.4), one has S(g, V)S(f, V) = S(fe, V) for morphisms / : IS -* mS and g : mS -> nS in the algebraic theory S. Thus the right S-module V determines the functor S( ,V):Sop -» Set. This functor sends the finite coproducts mS in S to the products Vm in Set. In other words, it is a finite-product-preserving functor from Sop to Set. Conversely, suppose that a functor F : Sop -» Set preserves products. Let V be the set 1SF. Then mSF is the power Vm. Consider an S-morphism A: IS -* mS;i, *+xxsx + ••• +xmsm. Its image under the functor F is an m-ary operation \F : Vm -» V on the set K Consider the type T= (S X {1}) U {(0,0), ( +, 2)} of S-modules. For each operation F : V°T -» V of these S-morphisms under the contravariant functor F make V into a T-algebra. Moreover, V satisfies the identities for an S-module. As an example, consider the identity xx +jCi( —1) = 0. The composite of the S-morphisms + : IS -» 25; xx<-* xx+ x2 and 2S -» IS; xx -» xx, x2 >-* xx( — 1) is the S-morphism 0 : IS -» IS. The image of the composite under the functor F is the derived operation xx + x , ( - l ) on V, which thus

ALGEBRAIC THEORIES

325

coincides with the scalar multiplication 0 on V. Finally, one has S(mS, V) = Vm = mSF for natural numbers m, yielding a natural isomorphism between the functors S( , IS 5 ) and F. Summarizing, S-modules V correspond exactly to product-preserving functors S op -+ Set, via the assignments V >-* S( K)and F ~ 15 f . Now consider an S-module homomorphism 0 : V -» W. One obtains a natural transformation 6 : S( , V) -> S( , W) whose component at the S-object mS is 6mS : S(mS, V) -» S(mS, W); v <-* vd. At an 5-morphism [i.e. (/ X m)-matrix] / : IS -» mS, the commuting of the diagram (HI 1.4.2) follows as vS(f, V)e,s = (fv)6 = f(v9) = u0mSS(f, W). Conversely, consider a natural transformation T:S( ,V)-* S( ,W). For each S-morphism A : 15 -» mS, the naturality (III 1.4.2) of T yields the commuting diagram S(mS,V) (3.4)

mS

S(\,W)

S(\,V) 5(15,V)

» S(mS,W)

is

-—.S{\S,W)

of functions. Setting A : 15 -> mS to be the insertion i( : 15 -* mS; xx >-* x¡ for each i in {1,...,m), one obtains 5(A,V) and 5(A,W) in (3.4) as the respective projections tri : Vm -* V and tr¡ : Wm -* W. Thus TmS is the product (T] S , . . . , TIS) of m copies of TIS : V -» If. Now considering general A : 15-*m5, yielding 5(A, V) and 5(A, W) as the corresponding m-ary operations (3.3) on V and W, the diagram (3.4) shows that riS : V -* W is an S-module homomorphism. In summary, one obtains the following: Theorem 3.1. Let S be the algebraic theory of a unital ring. Let 5 be the full s subcategory of the functor category Setsap iwhose objects preserve finite products Then 5 is isomorphic to the category Mod y . Proof. For F in Sj¡, define (3.5)

ilF = {\F:mSF -> 1 5 f | m e IM, A e 5(15, m5)}.

The isomorphism between 5 and Mod«, is then given by the functors with morphism parts (3.6)

(T : F - F') ~ [T 1S : {ISF, ÜF) -» (15 f ', ilF,)]

326

CHAPTER rv

UNIVERSAL ALGEBRA

(3.7) {d:V-+ W) -» [e-.So-'SetjjmS-* (S(mS,V)

-+S(mS,W);v

~> v6)].

Note that the isomorphism commutes with the forgetful functors Mod s -* Set and 5-» Set;F-> 1SF. D EXERCISES 3A. Verify that the r-algebra V = 1SF satisfies the 5-module identities JC,( - 1 ) + (*] + x2) = x2 and (x,s)i = x^st) for s,t e 5. 3B. Consider the type a = (5 x {1}) U {(0,0), ( - , 2)} of 5-modules. Show that each product-preserving functor F:S^> Set yields a cr-algebra structure on the set 15 f . Show that the cr-algebra 15 f satisfies the identity xx - (x2 — x3) = (xj - x 2 ) - * 3 ( - l ) . 3C. For an 5-module homomorphism 9 : V -» W, verify the naturality (III 1.4.2) of the transformation 6 : 5 ( _ , V) -> 5 ( _ , if) with components 6mS : S(mS, V) -» 5(n5, K); v -> Ü0 at each natural number w. 3D. Let SI be the small full subcategory of the category SI of semilattices whose objects are of the form «SI = (&{xx, ...,xn] - {0}, U ) for natural numbers n. (a) Show that nSl is the coproduct in SI of n copies of the object 1S1 = ({{*i)}, U ), for each natural number n. (b) Show that each semilattice (H, +) may be identified with S1(1S1, H) via S1QS1, H) -> H;6~* {xt}e (cf. Exercise II 2A and Section II 2.3). (c) For each semilattice ( / / , + ) and natural number n, show that there is a natural isomorphism H" -» Sl(/zSl, H); (hu..., hn) -» ({*,} -» ht). (d) Show that a semilattice H yields a product-preserving functor S l ( _ , H): Slop -> Set with morphism part ( / : / S l - m S l ) ~ (Sl(/,i/):Sl(mSl,/f) -* S1(/S1, H); h *+fh) [cf. (3.2)]. (e) Show that a product-preserving functor F : Slop -» Set yields a semilattice (1S1F, +) with join (1S1 -» 2S1; {*,} -» {xv x2})F. ci op

(f) Show that the full subcategory of the functor category Set whose objects preserve products is isomorphic to the category SI of semilattices.

ALGEBRAIC THEORIES

327

3.1. Set-Valued Functors The algebraic theory 5 of a unital ring is a small category. Theorem 3.1 identifies modules over the ring with certain functors from £ op to Set. namely those that preserve products. In preparation for the study of these and more general algebraic theories, as well as for other purposes, this section and the next consider the relationship between a small category V and the large functor category (3.1.1)

V ■=§£*"*.

The objects of V are sometimes described as presheaves over the category V. Note that the large category V is locally small For objects F : Kop -» Set and Q : y°v _> Set of V, the morphism class V(F, G) is the class of natural transformations T :F -* G. Such a natural transformation is given by the function T : V0 -> U AeV §et( AF, AG) of (III 1.4.1). Thus the class V(F, G) is a subset of the set Set(F0, UA£y0 Set(^F, AG)). It is convenient to establish some notation. The arrow in F o p corresponding to an arrow / : A0 -» Ai in V will occasionally be written as fop : At -» A0. The objects of V are covariant functors from F o p to Set, but they will sometimes be considered as contravariant functors from V to Sjt. Given such a contravariant functor F : V -* Set, the image of a F-morphism / : A -* B will be written as (3.1.2)

fF:BF-*AF;x~*fF(x).

In other words, the image function fF is written on the left of its argument. This has the advantage that, given a second F-morphism g: B -* C and an element y of CF, one may write fgF{y\ = (fg)F(y) = fF • gF(y) = fF(gF(y)). The connection between V and V is given by the (covariant!) Yoneda functor 3:V^V;(g:A->B)~(V(_,g):V(_,A)-+

F ( _ , B)).

Thus for an object A of V, the contravariant functor A3 from V to Set has morphism part A3:(f:A0

-> AJ ~ (fii3:V(AltA)

~ V(A0,A);

h ~ fh).

Using the notation (3.1.2), one has fAB(h)=fh for A in V(AV A). Given a morphism g : A -» 5 in F, the natural transformation G 3 : A3 -* B3 has component

at an object A0 of F o p . The naturality (III 1.4.2) of g3 at a Fop-morphism

328

CHAPTER IV UNIVERSAL ALGEBRA

fop : Ax -» A0 is the commuting of the diag;ram AA* (3.1.3)

8^A{

+

A»a

fA*

/ M ■

AA3 "™0

'

S3A0 ' A0

Consider an element h of AA3 = F U , , ^)- Then fAHhXg3At) = (#)g = /(/ig) = fBB(h(g3A )) as required: The naturality of g3, i.e. the commutativity of (3.1.3), reduces to the associativity (HI 1.4) in the category V. The crucial property of the Yoneda functor 3 : V -» V is described by the following: A

Yoneda Lemma 3.1.1. Given objects B of V and F of V, there are mutually inverse bijections V(B3,F)->BF;T~1BTB

and BF-* V(B3,F);x~>

(A ~>(V(A,B)

-AF;f

**fF(x))).

a As an immediate application of the Yoneda Lemma, one obtains Theorem 3.1.2. The Yoneda functor 3 : V -» V is a full and faithful embedding ofVinV. Proof. For objects B and C of V, the Yoneda Lemma yields the bijection V(B3, C3) s BC3 = V(B, C) on setting F = C3. D In the light of Theorem 3.1.2, the Yoneda functor 3 : V -* V is usually described as the Yoneda embedding of V in V. Objects of the image V3 are described as (contravariant) representable functors. For an object A of V, the representable functor A3 = F( , A) is represented by /I. Example 3.1.3 (Monoids). Let M be a monoid, realized as a small category according to Example III 1.2. The object part 3:{1} -» M0; 1 -* 13 of the Yoneda embedding yields the left regular representation 13:Mi op ->Setj;m -► ( m 1 3 : M ( l , l ) -*M(l,l);h

-» m/i)

ALGEBRAIC THEORIES

329

of M, while the morphism part 3:M(1,1) - M , ; m ~ ( m 3 : 1 3 -» 13) yields the right regular representation m3j : M(l, 1) - Af(l,

\);h~hm

of M [recall M(l, 1) = l 1 3 ]. As in Example III 1.4.2, the right actions are homomorphisms of the left actions. D The Yoneda embedding is a functor 3 : V -* V. Its object part sends objects to functors, and its morphism part sends morphisms to natural transformations. A functor ¡V:V -* Cat will now be constructed. Its object part sends functors to categories, and its morphism part sends natural transformations to functors: V -i* object -» morphism >-»

V functor natural transformation

-^» Cat -* category -» functor

Together, the functors 3 and fv connect all the fundamental concepts of category theory introduced in Section III 1. For an object F of V, i.e. a contravariant functor F : V -» Set, the small category ¡v F has (3.1.4)

( / > ) ={(A,x)\AeV0,x<=AF}

as its set of objects. For objects (A, x) and (B, y) of fyF, the morphisms f:(A, x) -» (B, y) are elements of (3.1.5)

[fFy(A,x),(B,y))

= {f<=V(A,B)\fF(y)=x}.

It is often convenient to write them in the form / : (A, x = fF(y)) -» (5, y). The composition of / : (A, x = fF(y)) -» (B, y) with g : (B, y = gF(z)) -+ (C, z) is JS:(A,x=fFogF(z))^(C,z). Note that this is well defined by the contravariance of F. The identity at (A, x) is 1A : (A, x) -* (A, x). Now consider a morphism T : F -* G in V, a natural transformation between the functors F and G. The functor ¡VT has object part / T : ( / F )

-(/G);(.4,;0~(>1,*T,).

330

CHAPTER rv UNIVERSAL ALGEBRA

Note x G AF =» XTA e AG. The morphism part of fvr is fr:(f:(A,x=fF(y)) v

- (B,y)) -

well defined since JTT^ = fFiy)rA A

(/:(¿,*T¿

= fG(yTB)

= / G (yr B )) - (B.yr,)),

by the naturality of T : F -* G.

The functoriality of fy:V -* Cat is readily verified (Exercise 3.1D). For an object F of V, the small category ¡v F is called the category of elements of F. Note that it comes equipped with the projection junctor

(3.1.6) PF : / F - F; (/: ( ¿ , x = / ' ( > ) ) - (B, y)) ~ ( / : ^ - B) v

to the original category V. For an object A of F, (3.1.4) specializes to (fyA3)0 = {(^4 0 'i) Mo G ^o» ^ e V(A0, A)}. One may identify (fvA3)0 with d,_1M}, the set of F-morphisms with codomain A, via the bijection d f ' M ) -» (/^.43) 0 ;g •-* (gd0,g). Composing 3 with fv, one thus obtains a natural underlying set functor U : V -» Set with morphism part

(3.1.7) l / í K - S e t í í / ^ - B ) - » ^ , - 1 ^ }

-»dr'Wíí-íf).

The faithfulness / = l^/17 of this functor yields Cayley's Theorem 3.1.4. Each small category is concrete. O Example 3.1.5. If V is a poset category, then (3.1.7) is essentially the down-set functor of Example III 1.2.3. D EXERCISES 3.1A. Verify the functoriality of the Yoneda functor, and prove that its object part injects. 3.1B. Prove the Yoneda Lemma 3.1.1. 3.1C. Let M be a monoid, considered as a left Ai-set (A/, Afop) via the left regular representation. Show that each A/op-homomorphism e : (Af, A/op) -* (A/, M° p ) is of the form R(m) : M -► M; x -» xm for an element m of M. 3.1D. Verify that fv:V-* Cat is a functor. 3.1E. Let Af be a monoid realized as a small category according to Example III 1.2. Show that Cayley's Theorem 3.1.4 specializes to Theorem O 4.3. 3.1F. Give an example of a small reinforced concrete category (V,G:V -* Set) for which the underlying set functor is not naturally isomorphic to the functor U: V -> Set of (3.1.7).

ALGEBRAIC THEORIES

331

3.1G. Let S be the algebraic theory of a unital ring, with 5 as in Theorem 3.1. Show that the Yoneda embedding 3 : S -» S corestricts to 3 : 5 -» 5, i.e. that the image 5 3 is a subcategory of 5. 3.1H. For each object A of V, show that (A, 1A) is a terminal object of fvA3. 3.2. Limits of Functors Let F be a small category. As shown in the previous section, the Yoneda embedding embeds V into the category V of contravariant set-valued functors on V. This section shows that the Yoneda embedding is the universal functor from V into a locally small, cocomplete category. First (Corollary 3.2.4 below), it will be shown that V is bicomplete. The proof to be given uses Currying for categories and functors that generalizes Currying for sets and functions (Example III 3.1.2 and references from there). Given small categories C and D, the product category C X D has object set C0 x D0, morphism set Cx X Dv identity function e : C0 x DQ -* Cx x D\, (x, y) -* (lx, ly), domain dQ and codomain dx functions d¡: Cx X D, -* C0 x D0; (/, g) <-* (fd¡, gd¡\ and composition (C X Z>XU0> ^oX^i» ^i)) x (C x DX(JC„ yO, (x 2 , y2)) -* (C X DX(x0, y0), {x2, y2))\ ((/„, g0), (/„ gl)) ~ (/o/j, goSO- I n other words, the category structure required by Definition III 1.1 is imposed componentwise on C X D. (Cf. Exercise 3.2A.) Note that the switching functor S.CxD^DxC with morphism part S:(f,g)~(g,f)

(3.2.1) is an isomorphism.

Theorem 3.2.1 (Currying Functors). Let E be a category, and let C and D be small categories. Then the functor categories ECxD and (ED)C are isomorphic. Proof. In the isomorphism, a functor F :C -» ED corresponds to a functor F :C X D -» E (the functor F "evaluated pointwise"). The morphism part of F sends a (C X £>)-morphism (/, g) : (x, v) -* (*', / ) to the composite y xF

(fF)

y

F > yx'F 8*' * y,x'F

of the component of the natural transformation fF :xF -* x'F at y with the image of the D-morphism g:y -* y' under the morphism part of the functor x'F : D -» E. In the inverse of the isomorphism, an arbitrary functor F:C X D -» E corresponds to the functor F:C -* ED whose object part sends a C-object x' to the functor x'F : D -* E with morphism part (g : y -» y') -» ((1,., g)F : (x', y)F -* (x', y')F). The morphism part of F : C -» ED sends a C-morphism / : x -> x' to the natural transformation fF:xF^> x'F with

332

CHAPTER IV UNIVERSAL ALGEBRA

component (fF)y = (/, ly)F at an object y of D. One may readily check that these object parts of the isomorphism and its putative inverse really are mutually inverse. The verification [Exercise 3.2B(a)] depends on the factorization (/, g) = (/, l y Xly, g) of a typical (C x Z))-morphism (/, g); (x, y) -» (x',y'). Now consider ECxD-objects F and G, with corresponding (ED)Cobjects F and G. Under the bijection ECxD(F,G)s(ED)C(F,G),

(3.2.2)

an £ CxD -morphism f : F -» G with component r(x ) : (,x, y)F -» (A:, y)G at an object (x, y) of C x D corresponds to an (£D)c-morphism T : F -» G with component (3.2.3)

TX:(XF:D-ȣ)

-»(xG : / ) - » £ )

at an object x of C. The component (TX)J, : yxF -» y i G of (3.2.3) at an object y of D is just T(X y) : (x, y)F -> (*, y)G. D Corollary 3.2.2. Let E be a category, and let C and D be small categories. Then the functor categories (EC)D and (ED)C are isomorphic. Proof. There is an isomorphism of EDxC with £ C x D whose object part sends a functor F : D x C -* E to its composite SF :C X D -* E with the switching functor S of (3.2.1). Theorem 3.2.1 then yields isomorphisms (EC)D s EDXC s ECXD = (ED)C. D The isomorphisms of Theorem 3.2.1 and Corollary 3.2.2 may be displayed by their morphism parts as follows: (EDf

(3.2.4) (T

» ECxD

* (EC)D,

: G - F) - (f : G - F) - (f : G - F),

extending the notation used in the proof of Theorem 3.2.1. For example, one has yxF = (x, y)F = xyF in E0 for x in C0 and y in D0. For g : y -* y' in Z)„ one has gxF = (gF)x. This notation is useful in the statement and proof of the Pointwise Limit Theorem 3.23. Let E be a category, and let C and D be small categories. Let F : C -* ED be a functor. (a) IfyF :C-*E has a limit TTyF for each object y ofD0, then F has a limit object bjn F : D -+ E;(g:y

- » / ) -> ()¿m(iryFgF) :üjnyF -* h j n / F ) .

ALGEBRAIC THEORIES

333

(b) IfyF : C -» E has a colimit tyF for each object y of D0, then F has a colimit object lira F : D -> E; (g : y -> y') ~> (lim (gfiy'f) : lngyF -* ljmy'F). Froo/ (a) Let A : ED -> (£ D ) C be the diagonal functor. Let r : ED -* E be the functor whose object part is B -» ((JE, y) -» y5), and whose morphism part is (

F yielding a terminal object of EDA/F. In other words, given an object

5A

(hm
(3.2.5)

(ümF)A 7T

of (EDA/FX
v(Um
^-

>(hmF)r

(3.2.6)

in ECxD. Thus it suffices to exhibit a unique element (3.2.6) of the morphism class (r, FX

(hm^)V (hjnyF)V yF TT

yF

yF

in Ec, the top row being specified uniquely by the terminality of iryF in

334

CHAPTER IV UNIVERSAL ALGEBRA

CV/yF. Thus (3.2.6) indeed exhibits a unique element of (I\ FX«p. ñF), namely with irFx > = TT¡F for x in C0, and with (hjn , = Jim £ r (b) Dual to (à). D Corollary 3.2.4. For a small category V, the category V = Set v" is bicomplete. y op

Proof. Consider a functor F :J -» Set with small domain. Now Set is bicomplete. Thus for each object A of V, the functor AF : J -» Set has a limit and colimit. D The universality of the Yoneda embedding amongst functors from V to locally small, cocomplete categories follows from the Extension Theorem 3.2.5. Each functor S:V -> C with locally small, cocomplete codomain extends via the Yoneda embedding to a cocontinuous functor S:V-*

C;(T:F-»

G) ~ (^((A,x)

~

- (ljmPG5)),

H^„A)):(^PPS)

unique up to natural isomorphism, that is left adjoint to the functor S:C ^ V;X~

((f:A

-+B) ~(C(BS,X)

-*C(AS,X);h

~fsh)).

Proof. The natural isomorphism C(FS,Y) = V(F,YS) required by (III 3.1.1) for the adjointness is given by the mutually inverse bijections ip's-JtA

C(FS,Y) - V(F,YS);k~ (using the diagonal functor A : C -* ClyF ) and

V(F,YS) -+ C(FS,Y);r>-* ljnj ((A,x)

**XTA).

The uniqueness of S follows by Exercise III 3.11, and its cocontinuity follows by Theorem III 3.4.1(b). D Corollary 3.2.6. Each functor F in Vis a colimit F s ln$(PF3:(A,x)

~ A3)

of representable junctors. Proof. ^Apply the Extension Theorem 3.2.5 to the Yoneda embedding 3 : V -* V. By the Yoneda Lemma 3.1.1, the right adjoint 3 : V -> V; F -» (A * V(A3, F))

ALGEBRAIC THEORIES

335

is isomorphic to the identity functor

ly-.V-*V;F~(A~AF). Thus the left adjoint 3 : V - V; F -» ljm PF3 is also isomorphic to the identity functor. D Corollary 3.2.6 clarifies the way a functor S : V -* C with locally small, A

A

cocomplete codomain extends to S:V -* C. For an object A of V, one has (3.2.7)

A3S = ljm \PA3S: ¡A3

-> c) = AS

by Exercise 3.1H and Exercise HI 2.50(a). For an object F of V, the cocontinuity of 5 then yields (3.2.8)

FS = (ljmF F 3)S = ljm (PF3S) = ljm (P F S).

EXERCISES 3.2A. Consider the forgetful functor [/:Cat-»Set whose morphism part sends a functor F :C -* D to its morphism part F :CX -» D,. Show that U creates products. 3.2B. (a) Verify that the respective object parts F -» F and F •-» F of the functors (F D ) C - F C x D and ECxD -» ( £ D ) C of the proof of Theorem 3.2.1 are indeed mutually inverse. (b) Verify that (3.2.2) is a bijection. 3.2C. Suppose that the limit of the functor F:C -» ED of Theorem 3.2.3(a) exists. For x in C0, specify the projection irf : hjn F -» ATF. 3.2D. Write out a proof of Theorem 3.2.3(b). In particular, do the dual of Exercise 3.2C. 3.2E. Verify that 5 and S in Theorem 3.2.5 really are functors. In particular, for an object Y of C, specify the morphism part of the contravariant functor YS.V-* Set. 3.2F. Let R be a unital ring. Show that each Ä-module is a colimit of finitely generated free Ä-modules. (Hint: Use Exercise 3.1G.)

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CHAPTER IV UNIVERSAL ALGEBRA

3.3. Finitary Algebraic Theories Theorem 3.1 described modules over a ring in terms of the algebraic theory of the ring (the small category of linear combinations and matrices over the ring) and in terms offinite-product-preservingpresheaves over the algebraic theory of the ring. With the preparation furnished by the previous two sections, the general version of this description may be presented. It offers the approach to universal algebra that extends the treatment of M-sets over a monoid M as functors from the category M to the category of sets. Definition 3.3.1. A (finitary) algebraic theory V is a small category whose object set V0 = {nV\n e N} contains, for each natural number n, the coproduct nV of n copies of the object IV. D Example 3.3.2. Let K be a prevariety of algebras of given finitary type T : Ü -* N. For each natural number n, let nK be the free ^-algebra on the subset {xu...,xn) of (1.3.4). Note that the set {xu...,xj is the disjoint union or coproduct of n copies of the set {JCJ. Since the free algebra functor ílRg-.Set-* K is cocontinuous, being left adjoint to the underlying set functor U:K-* Set, it follows that nK is the coproduct in K of n copies of IK. Let K be the small full subcategory of K with object set {nK\n e N). Then K is a finitary algebraic theory, the algebraic theory of the prevariety K. D

The general version of Theorem 3.1 is obtained by reversing the construction of Example 3.3.2. Let V be an algebraic theory. Define the type rv of V to be the disjoint union (3.3.1)

rK= U

[V(lV,nV)^{n}],

with operator domain ilv = ^,^NV(IV, nV). Given the insertions t, : IV -* IV for 1 £ i £ /, an arbitrary F-morphism f:lV-*mV is described as the coproduct (3.3.2)

/=

£/, i-i

with f¡ = t , / G Ty1^). Thus the operators from Clv yield the full set Vx of morphisms of the small category V via (3.3.2). A model of the algebraic theory F is a (covariant) functor A : Kop -» Set that preserves finite products. It is convenient to denote the image of IV under the functor A by the same symbol A, i.e. using A for the set IVA.

ALGEBRAIC THEORIES

337

Since A preserves finite products, one then has mVA =Am for natural numbers m. An element a> : IV -» mV of r"yX{m] thus yields the m-ary operation
(3-3.3)

VI si */,(£), = / J £ j

in Ty^n) by (III 2.1.3). Applying the covariant functor A : Vop -» Set to (3.3.3), one obtains m

(3.3.4)

^^i^lÁf8)t=f^Y\gfy'-i

In other words, the r^-algebra (A,ilv)

satisfies the identities

(3.3.5) ( £ ) , ( * ! , . . . , xn) =fi(gl(x1,,..,!,)

£„,(*,,..., xn))

for each composable pair (f:lV-* mV, g : mV -* nV) of F-morphisms. The variety of T ^algebras satisfying all such identities is called the variety V of the algebraic theory V. Let ModK denote the full subcategory of V consisting of models of V. The general version of Theorem 3.1 may then be formulated. Theorem 3.3.3. Let Vbe afinitaryalgebraic theory. Then the categories Vand Mod y are isomorphic. Proof. A model A : Vop -» Set has already been observed to yield a K-algebra (A,ftK).Conversely, given a F-algebra (A, ilv), one obtains a functor A:Vop -» Set with morphism part (3.3.6)

lZfl:lV-"nv\~(A°'-+A,;(al,...,am)

~(/f>i

aj,...,//V„..-,«„,)))•

The functoriality of A : Kop -» Set follows from the satisfaction of the identities (3.3.5). Homomorphisms 9 : (A,fiK)-» (B, dy) of F-algebras correspond to the components at IF of natural transformations
338

CHAPTER IV UNIVERSAL ALGEBRA

Conversely, for a natural transformation 6 : A -» B, the naturality A" —-^

Bn *¡

at the insertion i, : IV -> nV for each i guarantees that the function 6nV : A" -> B" has the form A" -» B"; (au..., an) -» (a,0 1 K ,..., a„01K). D By virtue of Theorem 3.3.3, the categories V and Mod,/ determined by the algebraic theory V will be identified. One consequence is the following: Corollary 3 3.4. The full subcategory VofV consisting of finite-product-preserving functors is (small) bicomplete. Proof. By Theorems 2.1.3 and 2.2.3, the variety Fis bicomplete.

D

Specializing Example 3.3.2, let K be a variety of algebras of given finitary type T :ft -+ N, with corresponding algebraic theory K. Let V be the variety of the algebraic theory K. Thus Fis a variety of algebras oftype rK : ftK -» Fd. For each element (w, n) of the closed operator domain fl, there are corresponding elements xl... x„(w, n) of nK and (w, n)K : IK -* nK; xx -» * , . . . xn(w, n)K of K(1K, nK). If u = u with n = max(({arg u) U (arg v)) is an identity satisfied by every ^-algebra, then (u,n)K = (v, n)K. Consider a F-algebra (A, SlK), identified with the model A_: K0? -» Set of K. Then the underlying set A (or \KA) becomes an^ ft-algebra under the action (w, n)£ : A" -» A of the element (w, n) of fi. By reduction, one obtains the algebra (A, ft) of type T : ft -» N. Since (w, n)£ = (i>, n)j£, the r-algebra (/l, Ü) satisfies each identity u = v of K. Thus the F-algebra (A, ftK) yields a ^-algebra (A, ft). Conversely, consider a X-algebra (A, il). For each natural number /, let r\, be the component at {*,,..., x,} of the unit TJ of the adjunction (ftRK, U, TJ, e) determined by the (pre-)variety K. Then (A, ft) determines a model A : K09 -» Set with morphism part (3.3.7)

(f:IK->mK)~(s*({xlt...,xm),A) - Set({*i

x,),A)\a~

^[/(anj^f)-

This model is identified (according to Theorem 3.3.3) with a K-algebra

U,ftKl

ALGEBRAIC THEORIES

339

The upshot of the correspondence between ^-algebras (A, ft) and (F-algebras or) íí-models (A,ílK) is that the Ä-models provide an "invariant" description of the algebras, independent of any arbitrary choice T : il -> M of a type for the algebras. This invariance is normally preferable due to its elegance and naturality. EXERCISES 3.3A. Determine the algebraic theory of each of the following prevarieties: (a) the variety Set of algebras of empty type; (b) for a monoid M, the variety M of Af-sets of type M X {1}; (c) the prevariety K of convex sets [as in Example 2.1.5(a)]; (d) the variety SI of semilattices of type {(•, 2)}. 3.3B. Consider the set P = {x¡, x2,...} of positive integers as in (1.3.4). Define a small subcategory Set2 of Set by Set20 =

{{xl,...,xn}2\n<=N)

and Set 2 ({*!,...,

JCM}2,{*„.

• •, x„}2)

= {(/, 8) 1/. g e Set({*„..., xm)\ {*,

*„})}

natural numbers m, n. Show that Set2 is an algebraic theory. Determine the variety Set of Set2. A rectangular band is a semigroup satisfying the identity xxx2x^ = xx x3. Show that Set is the variety of rectangular bands. 3.3C. Give a detailed verification of the functoriality of (3.3.6) in the proof of Theorem 3.3.3. 3.3D. Give a detailed verification of the functoriality of (3.3.7). 3.3E. Let F be a finitary algebraic theory. Let C be a locally small category, and let 5 : V -» C be a functor. For a C-object X, show that there is a for (a) (b) (c)

functor with object part C ( _ 5, X) : Vop -* Set; nV -» C(nVS, X). Show that C( S, X) is a presheaf over V that yields a K-algebra if S preserves finite coproducts.

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CHAPTER IV UNIVERSAL ALGEBRA

3.3F. Let V be the poset category with Hasse diagram 0 -» 1. (a) Show that V is an algebraic theory, with IV = 1. (b) Is V the algebraic theory of a prevariety? 3.4. Theory Maps The full power of the approach to universal algebra via algebraic theories becomes apparent when studying transformations between theories. Definition 3.4.1. Let V and W be finitary algebraic theories. Then a theory mapM: W -* V is a functor preserving finite coproducts such that nWM = nV for each natural number n. D Given a theory map M : W -* V, the composite with the Yoneda embedding 3V : V -» V yields a functor M3V : W -> V; nW -* V( , nV) with locally small, cocomplete codomain. By the Extension Theorem 3.2.5, there is then a unique extension M 3V or (3.4.1)

M:W^>V;F~>

li^\PFM3v:

jF ^ V\

that is left adjoint to the functor M 3V or M:V^>W; V(lV3y, F)) or

F -» (IW -»

(3.4.2) M:V->W:F~>

({f:lW^mW)

-» (fMF : mVF -> IVF)).

[Note the use of the Yoneda Lemma 3.1.1 to obtain the simple form (3.4.2) of the object part of M. Also see Exercise 3.4F(a).] Proposition 3.4.2. Given a theory mapM: W -» V, the right adjoint M : V -» W restricts to a functor (3.4.3)

Mr:V-+W

between the corresponding varieties. Proof. Let A : Vop -* Set be a F-model. Then for each natural number n, one has (3.4.4)

(nW)( AM) = nVA = (IVA)" = (1W)(AM)".

D

Corollary 3.4.3. The functor Mr of (3.4.3) commutes with the underlying set functors U : F-> Set; A -» IVA andU:W-* Set; 5 ^ 1WB.

ALGEBRAIC THEORIES

341

Proof. Set n = 1 in (3.4.4). D Starting from the prevariety Set of algebras of empty type, the construction of Example 3.3.2 yields the algebraic theory Set of Set. For any algebraic theory V, there is a theory map (3.4.5)

/ : Set -» V; ( i, : 1 Set -» n Set) -* ( i, : IV -> nV).

Those F-morphisms that lie in the image of / are described as combinatorial. By (3.4.2), the functor / ' : £ - » Set is the forgetful functor [/:F-> Set; (A, ily) -* A. Since this functor appears in the adjunction (ClvRv, U, TJ, el, it has a left adjoint ClyRv: Set - * V or / ' : Set-» V, the free algebîa functor for the variety V. TheorerrT 3.4.4 below constructs a left adjoint M' : W-* F to each functor Mr : V-* W given via (3.4.3) by a theory map M : W -» V. The construction is based on a study of the embedding of the variety V in V. It turns out that this embedding is analogous to the embedding of a prevariety K of r-algebras in the category T of all T-algebras. (In the terminology of Exercise III 3.2E, both embeddings are reflections.) For an algebraic theory V, let Ev : V -» F denote the embedding of F in F via the Yoneda embedding. By Corollary 3.3.4, the locally small category V is cocomplete. By the Extension Theorem 3.2.5, the embedding Ey:V-*V extends to a functor Ev : V -* V that is left adjoint to the embedding Ev :V~* V. The functor Ev : V -» V; F -» lim (PFEV : fvF -» £) is analogous to the replication functor RK : T-> K. In particular, note that the colimit of a functor F:I-*V_ differs from the colimit of the composite functor FÊy :I -* V, just as colimits of ^-algebras in K differ from their colimits in T. A theory map M:W -* V yields the following system of functors:

■+V J

Ey

M' M'

M

A

M M Ew

W —— Wt—r-* w E

w

—

Ew

Note the description of M' : V-* W as Mr = EVMEW by Proposition 3.4.2 and the definition of (3.4.6)

M' = ÈwMÊy.

*

.

342

CHAPTER IV UNIVERSAL ALGEBRA

Theorem 3.4.4. Given a theory map M:W' -» V, the functor MT : V-* W has the left adjoint M' : W-+ F; B ~ fog (PBMEV : fwB -* V) Proof. By composition of adjunctions (cf. Exercise III 3.1K), the functor MÊV : W -» V is left adjoint to ÈVM : V-> W. Thus for A in VQ and B in J£ 0 , one has V^BM1, A) = V¡.BEWÚEV, A) s # ( £ £ „ , , /1£ K M) = j£Cfl, AM'), as required. D Example 3.4.5. (a) (Replication.) Let X be a variety of r-algebras for a certain type nil -* N. The replica functor R^ : g-* K restricts to a theory map R:T^K between the corresponding algebraic theories determined as in Example 3.3.2. Then Rr : K-* g is just the inclusion, while R' : g-* K is the replica functor. In other words, the adjunction of Theorem 3.4.4 yields the adjunction of Theorem 2.1.3 in this case. (b) (Free Algebras.) Taking the theory map / : Set -» V of (3.4.5), Theorem 3.4.4 builds the free K-algebra XJ' over a set X as a colimit of finitely generated free algebras. (c) (Geometric Realizations.) Let K be the algebraic theory of convex sets [cf. Exercise 3.3A(c)]. Let R be the algebraic theory of the ring of real numbers, as in Section 3. For each natural number n, the embedding of the (n — l)-simplex nK in nR (cf. Exercise 2. ID) yields a theory map R : K -* R. Then Rr : R-* K construes real vector spaces as convex sets. Its left adjoint R1 : K-> R builds the so-called geometric realization of convex sets (and other members of the variety | p . (Note that the variety K properly contains the prevariety of convex sets—cf. Exercise 3.4G.) (d) (Knot Groups.) Let Kw be the variety of quandles, right quasigroups satisfying the identities x1xl = xx of idempotence and *jjt3 • x2x3 — xxx2 -x^ of distributivity. Let M : Kw -* Gp be the theory map given by (JC, -» xx -x2) -» (*i -» x2~xxlx2) and (x, -^X,/JC 2 ) -» (xt -» x2xlx2 x). Thus for a group G, the quandle GMr has multiplication xx -x2 =x21x1x2 and right division xx/x2 =x2x1x21. For a quandle Q, the group QM' is the group Adconj Q constructed by D. Joyce ["A Classifying Invariant of Knots, the Knot Quandle," /. Pure Appl. Algebra 23 (1982), 37-65]. Joyce showed how to associate a quandle Q(K) to a knot K such that Adconj Q(K) becomes the knot group irjlS3 - K), the fundamental group of the complement of the knot in S3. Joyce's construction is a precise version of a method outlined by W. Wirtinger at the 1905 meeting of the German Mathematical Association in Meran ("Über die Verzweigungen bei Funktionen von zwei Veränderlichen," Jahresbericht der DMV 14 (1905), 517.) D The applicability of Theorem 3.4.4 depends on recognizing which functors R : V-» W are of the form R = Mr for a theory map M : V -* W. To this end, the following characterization of right adjoints with codomain W proves useful. Consider a category C with finite coproducts. For a naturaTnumber n, write nZ for the coproduct of n copies of an object Z of C. A W-coalgebra in C is defined (via Definition 2.4.4) as an object Z of C op ® ¡V. Thus an

ALGEBRAIC THEORIES

343

n-ary operator a> of W yields a cooperation : Z -* nZ on Z. If C is locally small, then (asTñ Exercise 2.4L) one has that Z is an object of C ^ ® ^ iff, for each X in C0 = C0°P, the set C i - Y , Z) - C(Z, X) lies in i£ Theorem 3.4.6. Let W be an algebraic theory. Let C be a locally small, cocomplete category. Then a functor R:C -* ¡Vis a right adjoint iff there is a W-coalgebra Z in C such that R:C -* W is naturally isomorphic to C -* ¡V; X~C(Z,X). Proof. If R has a left adjoint 5, then (3.4.7)

C(1WS, X) s W(\W, XR) = XR

for X in C0. On the one hand, (3.4.7) shows that one may take Z = 1WS to obtain the required form for R. On the other hand, (3.4.7) shows that \WS is a H£-coalgebra in C. Conversely, consider the functor R : C -» W, X >-» C(Z, X) for a |£-coalgebra Z in C. Define (3.4.8)

S:W^>C;(a>:lW^>nW)

-» (w z : Z -* nZ).

Note that 5 : W -* C preserves finite coproducts. By the Extension Theorem 3.2.5, the functor S:W-> C extends to a functor 5 : ^ - * C that is left adjoint to the functor S:C ^> W;X~ (nW* C(nWS, X)). Now S : C -* ^ corestricts to the functor J?: C -» | £ (cf. Exercise 3.3E). Thus for X in C0 and y in Wp, one has CiKS, Z ) = W(Y, XS) = ¿£(7, XR). In other words, the restriction S : W^> C to W of 5 : W -» C is the required left adjoint. D Corollary 3.4.7. Let M:W-* Vbe a theory map. Then \V is a W-coalgebra = inV. Proof. Apply (3.4.7) with (/? : C -» # ) = (Mr : V-> W) and S = M'. Thus ¿(IF, * ) = H1WS, * ) s mXW, XR)sXReWp for * in £,. D Theorem 3.4.6 serves to characterize functors R : V-> W that appear as ' '■£-* EfoT a t h e o r v m a P M\W-*V.

M

Corollary 3.4.8. Let V and W be algebraic theories, yielding underlying set functors Uv or U:V-> Set; A -► IVA and Uw or U:W-> Set; B -» ÍWB. Then a functor R:V-* WisoftheformMr:V-*WforatheorymapM:W-* V {and so possesses a left adjoint) iff it commutes with the underlying set functors: Uy = RUW. Proof. The "only i f direction is Corollary 3.4.3. Conversely, suppose Uv = RUW. Then for each K-algebra X, the set V(1V, X), naturally isomorphic with XU and XRU, carries the Wf-algebra structure XR. Thus IV is a

344

CHAPTER IV UNIVERSAL ALGEBRA

|K-coalgebra such that R : C -» W takes the form X -» V{\V, X). The required theory map is then defined by (3.4.8) with C-V and Z-W. D Example 3.4.9. Let K be a unital, commutative ring. As in Exercise 2.2H, let CAlgy be the variety of commutative /i-algebras. Recall that coproducts in CAlgy are given by the tensor product construction, and that the polynomial algebra K[xt] is the free commutative K-algebra lCAlg^ on {x¡}. The reduction functor R : CAlgy -» Mon forgetting the if-module structure preserves underlying sets, and is thus of the form Mr : CAlgy -» Mon for a theory map M : Mon -» CMgK by Corollary 3.4.8. By Corollary 3.4.7, it follows that K[x¡] is a comonoid in CAlgr. The comultiplication is given by the /C-algebra homomorphism (3.4.9)

S : K[xx] - K[xr] ® K[x,]; xx ~> xx ® 1 + 1 ® x„

while the counit is given by e : K[xx] -» K; x, -» 0. Objects of Mon ® CAlg^ such as #[*,] are called Hopf algebras. D

EXERCISES 3.4A. For an algebraic theory V, verify that the functor Ev : V-* V is just the inclusion of the category of F-models in the category of presheaves over V. 3.4B. Show that each homomorphism H: R -» S of unital rings yields a theory map between the corresponding algebraic theories. Determine Hr : Mode -> Mod» and H' : Mod., -» Mod,. 3.4C. Show that the inclusion H : U -» C yields / / ' : ModD -» Modo as the complexification of Exercise III 3.6L. 3.4D. Let B be a subgroup of a group A Consider the varieties B of B-sets and A_ of /I-sets. Denote the corresponding algebraic theories by B and A. Show that the inclusion of B in ^4 yields a theory map M:B -* A such that the adjunction of Theorem 3.4.4 between M' and Mr becomes the adjunction (III 3.1). 3.4E. Let V and ¡V be varieties of algebras of given type T : ÍÍ ~* N such that each K-algebra is a ^-algebra. Show that the replica functor R v : W-* V restricts to a theory map R : W -* V. Determine the adjoint=functors R':W-> Fand Rr:V-+W. 3.4F. Let M : W -» V be a theory map. (a) Show that the morphism part of M : V -* W takes the form (T : F -» G) ►* (nW ~ (n^P - nFG; A: ~ XT,»,)).

ALGEBRAIC THEORIES

345

(b) Using Theorem 3.3.3, show that the morphism part of AT : V-* ¡V takes the form (e:(A,ily)

-> (B,ilv))

-* (0:(A,ñw)

-+

(B,Slw)).

Thus note that the non-triviality of M and M" is encapsulated in (3.4.2). 3.4G. Let K be the algebraic theory of convex sets, as in Example 3.4.5(c). (a) Let (H, • ) be a semilattice. Show that H yields a model of K with (p:\K-* 2K)H = (H2 -* H; (x, y) -* xy) for a tf-morphism p whose image is an interior point of the closed bounded real interval 2K. (b) Show that H is an element of K that is not a convex set if \H\ > 1. (c) Determine the geometric realization of the free semilattice 2S1 over a 2-element set. (Hint: Describe 2S1 as a coequalizer of convex sets, and use the cocontinuity of /?'.) 3.4H. Realize the free semilattice nSl on {JCJ,..., xn) as the poset of nonempty subsets of {xv...,xn}. Realize the free Boolean algebra nBAlg on {jtj xn] as the poset of all subsets of {xly..., xn}. (a) Show that inclusion yields a theory map M : SI -» BAlg. (b) Determine AT : BAlg-* SI and M1 : SJ.-> BAlg. In particular, identify the Boolean algebra {1 < 2 < 3}A/'. 3.41. Prove the following Extension Theorem for varieties: Let V be a variety, with corresponding algebraic theory V. Then each functor S:V -* C with locally small, cocomplete codomain extends to a unique cocontinuous functor S1 : V-> C with EyS' = 5. 3.4J. Let M : W -» V be a theory map, and let B be a ^-algebra. Using Corollary III 2.5.6 and Theorem 2.2.3, give an explicit construction of the F-algebra BM' as a quotient of a free F-algebra. 3.4K. Construe the functor MonT : Mon -> r of Exercise 1.20 as the right adjoint M' : Mon -» r determined by a theory map M : r -» Mon. What is the left adjoint M' : T-> Mon? 3.4L. Let T ¿ p - » N b e a finitary type, and let a : ^ -+ N be the restriction of T : SI -* N to a subset ^ of SI. Let R : T-> a be the functor sending a T-algebra {A, Si) to its {non-basic) reduct ^ 4 , ^ ) [cf. Section 1.3]. Show that the reduction functor R : g-> g has a uniquely defined left adjoint S : a-> T. The left adjoint is known as the augmentation functor adjoint to R. [Hint: Note that the "geometric realization" of Example 3.4.5(c) is an instance of such an augmentation functor.] 3.4M. Let R : Ring -» Mon be the (restriction of the) reduction functor forgetting the abelian group structure of a ring. (a) Show that i?:Ring-> Mon has a left adjoint S :Mon-* Ring. The left adjoint is known as the monoid ring functor.

346

CHAPTER IV UNIVERSAL ALGEBRA

(b) Show that the monoid ring functor sends the symmetric group 5 3 to the ring Z[S3] of Exercise III 2.2D. 3.4N. Verify the details of Example 3.4.9. In particular, show that the image of a power x" under the comultiplication (3.4.9) is E/Loí"!*" - ' ® x\3.40. Let n be a positive integer. Consider the functor Rn : ModT -» Mod7 with object part A *-» {x e A | nx = 0}, and with morphism part given by restriction. Use Theorem 3.4.6 to show that Rn has a left adjoint. Determine the left adjoint. 3.4P. Let G be a group. (a) Show that the trivial homomorphism G -» {1} yields a theory map M : G -* Set whose left adjoint Ml :Q_-> Set maps a G-set X to the orbit set X/G. (b) Show that the right adjoint A/ r :§et-» G is left adjoint to the functor Fix:G-» Set; (X,G) -» F i x ^ . G ) . 4. MONADS Let (M, •, e) be a monoid. As in Example III 1.2.2, the fixed set M yields a functor T : Set -» Set with morphism part (4.1) (f:Zl-*Z2)^((f,lM):ZlXM-*Z2xM;(z,m)^(zf,m)). However, the monoid structure on the set M associates extra structure with the functor T. For example, the multiplication in the monoid yields a natural transformation fi : T2 -» T whose component at a set X is (4.2)

fjLX : X X M X M ^> X X M; (x,mi,m2)

-»

(x,mim2).

(Cf. Exercise III 3E.) Note that (4.2) is the action of the monoid M on the free M-set X X M over the set X. The natural transformation n : T2 -> T satisfies the associative law (4.3)

(7)*)M*(M7>.

Indeed, the component at a set X of the left-hand side of (4.3) is

(Tfi)xfix

or

(x,ml,m2,m3)

—* (x,ml,m2m3)

•—* \x,mx

•m2m-i),

while the component of the right-hand side is ( fiT)x fix or T

(x, /Wj, m 2 , WÎ3) •—* (x, /Mj/n2) /n3) •——* (x, /nj/^.tf^).

MONADS

347

These components agree by the usual associative law (O 4.2) in the monoid M. The identity element e of the monoid yields a natural transformation Tj : l j ^ - * T whose component at a set X is (4.4)

(x,e).

TJX:X^XXM;X~

Using the identity transformation l r : T -* T, the left unit law (4.5)

(t?7>=lr

and right unit law (4.6)

(7V,)M-lr

are satisfied. For example, the left-hand side of (4.5) has component (j)T)xßx or (x,m) ■—* (x,e,m)

>—* (x,em),

while the component of the right-hand side is the identity function on Xx M. These components agree by the validity of the left hand side of (O 4.3) in the monoid M. Overall, the monoid structure on the set M is embodied in the triple (4.7)

(T,UL,V)

or monad in the category of sets, i.e. a functor 7*: Set-» Set with natural transformations i¿:T2 -* T and Tjrlg^-* T satisfying the associative law (4.3) and unit laws (4.5), and (4.6). EXERCISES 4A. Determine the component of the left-hand side of (4.6) at a set X, and verify that it agrees with the component of the right-hand side of (4.6) at X. 4B. Suppose that the monoid M is idempotent. Express its idempotence as a property of the natural transformation fi:T2 -* T. 4C. For the fixed monoid M, the functor T:Set-* Set of (4.1) has a right adjoint T : Set-* Set with object part X -» XM (cf. Example III 3.1.2). Does the monoid structure on the set M associate extra structure with the functor T"> [Hint: Dualize (4.7).] 4D. Let K be a commutative, unital ring, and let Y be a ^-algebra. For the functor SY of Section III 3.6, show that there are natural transformations ju.:SY-* SY and rj: lMod|c -» SY satisfying the associative law

348

CHAPTER IV UNIVERSAL ALGEBRA

(Sy/it)/x = (liSy)ti and unit laws (TjSy)/¿ = 1 = (SyTj)/x. The component of fi at a X-module Z is fiz: Z 8 Y 9 Y -* Z 9 Y; z 8 y 9 y' -» z 8 y / , while the component of TJ at Z is T J Z : Z ->Z

8 Y; z -> z 8 1.

4E. (a) Give an example of two monads (T, /¿, TJ) and (7\ /¿', TJ') in the category of sets for which \L ¥= ¡¿ and TJ = TJ'. (b) Give an example of two monads (T, ¡i, TJ) and (T, \¿, TJ') in the category of sets for which TJ # 17'. 4F. Let ( r , fi, e) be a monad in the category of sets. Show that the image under T of the terminal object of Set is a set carrying a monoid structure. 4.1. Monads and Their Algebras The functor T : Set-» Set; Z -» Z X M of (4.1) yielded the monad (7\ ¿1, TJ) of (4.7) that reflected the monoid structure (M, •, e) on the set M. The definition (4.7) may be extended to more general categories. Definition 4.1.1. A monad or triple (T, p, 17) on a category B is a functor T : B -* B equipped with natural transformations fi : T2 -» T (called composition) and TJ : 1B -» T (called the unit), such that the associative law (Tn)n = (/nD/u,, thefe/runir iflw (TJD/A = l r , and therig/zfunir /aw (TTJV = l r are satisfied. In this context, the category B is called the base category. D Example 4.1.2. (a) (Closure Operators.) A monad T on a poset category (B, :S) is called a closure operator. The existence of the unit yields V* e B, x <, xT. The existence of the composition yields V* G S, xTT <, xT. On the other hand, the monotonicity of the functor T applied to x <> xT yields xT ^ xTT. Thus xT = xTT for each ac in B. (b) (Ä-Algebras.) For an algebra Y over a commutative, unital ring K, the construction of Exercise 4D yields a monad (5 y , /u., 17) on the category Mody of íC-modules. D Given a Galois connection (5, /?, TJ, e) between functors S : ( 5 , ^ ) -» (/I, ^ ) and R:(A, &)-* (B, ^ ) , the composite functor T = SR yields a monad or closure operator on the poset (B, :s), as corroborated by (III 3.3.2Xa) and (III 3.3.3Xa). This phenomenon is quite general: Each adjunction yields a monad. Proposition 4.13. Let (F, G, TJ, E) be an adjunction between functors F : B -» ^4 anrf G : A -* B. Then (FG, FeG, TJ) « a monad on the base category B.

MONADS

349

Proof. For .Y in B0, the naturality of e : GF 1, at eyF : XFGF -> XF yields the commutativity of the diagram XFGFGF (4.1.1)

-XFGF

XFGF

-GF XF

-XF

B

XFGF

a

XF

■* XF

in A. Applying the functor G then yields exFGFeXF = exFGßxF> i-e- t n e equality of the respective components of (FG • FeGXFeG) and (FeG ■ FGXFeG) at X. Thus the associative law holds. Now by Proposition III 3.1.7, one has rjxeXF = 1XF. Applying the functor G yields r)x°exF = ^XFC uethe equality of the respective components of (-qFGXFeG) and l f G at X. Thus the left unit law holds. The right unit law follows similarly (Exercise 4.1 A). D Example 4.1.4 (Monad of a Prevariety). Let K be a prevariety of algebras of type T : fi -» IU According to Proposition 4.1.3, the adjunction (ílRp U, 17, e) of Section 2.1 then yields a monad (SIR^U, ClR^eU, 17) on the category Set, the so-called monad of the prevariety K. For instance, if K is the prevariety of A/-sets for a monoid M, the monad of K is the monad (4.7).

□

Proposition 4.1.3 shows how to obtain a monad from an adjunction. Conversely, each monad (,T : B -* B, ft, 17) yields an adjunction, constructed by analogy with the relationship between the adjunction (CIR^, U, -q, e) and the monad (fti? Ä i/, ClRKeU, TJ) of a variety K of algebras. Definition 4.1.5. Let T : B -» B be a functor on a base category B. (a) A T-algebra (X, a) is an object X of B equipped with a B-morphism ax or a : XT -* X, known as the action of T on X. (b) A T-homomorphism h : (X, a) -* (X', ci) or homomorphism of T-algebras is a 5-morphism h : X -* X' between T-algebras, such that the diagram XT (4.1.2)

hT X'T

-* X'

commutes. (c) The category B of T-algebras has all T-algebras as its objects and all homomorphisms between them as its morphisms. D

350

CHAPTER IV UNIVERSAL ALGEBRA

Definition 4.1.6. Let ¡x or (T : B -» B, ¡x, 17) be a monad on a base category B. (a) A p-algebra is a T-algebra ( X, a) such that the associative law, i.e. the commutativity of XT2 --i- 2 -» XT (4.1.3)

a

aT XT -

-

■ X

in B, and the «mi law, i.e. the commutativity of

(4.1.4)

in B, are both satisfied. (b) A fi-homomorphism h : (X, a) -» (A", a') or homomorphism of ^-algebras is a r-homomorphism between /¿-algebras. (c) The Eilenberg-Moore category ^B of the monad y. or category of ¡¿-algebras has all ¿¿-algebras as its objects and all homomorphisms between them as its morphisms. D The Eilenberg-Moore category of the monad of a prevariety K includes objects (XU, e%) for each ^-object X, the action being given by the image under U:K-* Set of the component at X of the counit e of the adjunction (ÜRK, U, 17, £)ÜCf. Exercise 4.1F.) The Eilenberg-Moore category of a monad is the domain for the right adjoint in an adjunction. This adjunction then returns the monad by the construction of Proposition 4.1.3. One thus obtains the following converse of Proposition 4.1.3. Eilenberg-Moore Theorem 4.1.7. Let \L or (T : B -* B, ¡x, 17) be a monad, with Eilenberg-Moore category *B. Then there is an adjunction (5 : B -»^B, R : Mß -» B, TJ, Ï ) , called the Eilenberg-Moore adjunction of the monad (T, fi, 17), such that (T, p, TJ) = (SR, S"eR, 17).

MONADS

351

Proof. The putative right adjoint R.^B -» B is the forgetful functor with object part {X, a) >-* X. The putative left adjoint S:B -»MB is defined to have morphism part (4.1.5)

(f:X' ^X) ~ (fT:(X'T,nx.) ■* (XT,ßx)).

Note that the component at X of the associativity of the monad (T, fi, -q) yields the associativity of the T-algebra (XT, fix). Similarly, the component at X of the left unit law of the monad (T, n, 17) yields the unit law in the T-algebra (AT, fix). The component at a ¿t-algebra (X, a) of the counit ë is defined as e(X a) = (a:(XT, ¡ix) -* (X,a)). Note that this component is indeed a homomorphism of 7-algebras, by the associativity of (X, a). Moreover, the naturality of ë at a ''ß-morphism h : (X, a) -» (X', oí) follows since the S-morphism h : X -» X' is a homomorphism of T-algebras. If X is an object of B, the component at X of the left unit law of the monad (T, /u,, r¡), namely TJX • ¡xx = \XT, yields the component T)X • exs = 1^5 at X of the identity (rjSXSê) = 1 of Proposition III 3.1.7. Similarly, if (X, a) is an object of ^B, then the unit law T)X ■ a = lx in the /¿-algebra (X, a) yields the component T\X a)R-~e*x tt) = \(X a)R at (X,a) of the identity (RrjXeR) = 1 of Proposition III 3.1.7. With its hypotheses satisfied, Proposition III 3.1.7 then shows that (S, R, r¡, ê) is an adjunction. Finally, the component at a 5-object X of the natural transformation SêR is 5* s = ^..Mjr) = ( Vx • (XT2, M*r) ~» (XT, /**))* = (nx-XT2-* XT), so that SëR = ß, as required. D Example 4.1.8. Let K be a commutative unital ring, and let Y be a üf-algebra. Let (SY, M» *f) D e t n e monad in ModK given by Exercise 4D and Example 4.1.2(b). Then the Eilenberg-Moore category MMod^ is the category of Y-modules (Exercise 4.1G). D

EXERCISES 4.1A. Complete the verification of the right unit law in the proof of Proposition 4.1.3. 4.1B. For a monoid M, verify that the monad of the variety M of Af-sets is the monad (4.7). 4.1C. Let A be a set, and let T be a closure operator on the power set (&(A\ ç). The operator T is said to be topological if 0 T = 0 and V*. Y çA,(X U Y) r = XT U YT. Show that the set {X çA | X = * r } of fixed points of a topological closure operator T is the set of closed subsets of A in a topology on A.

352

CHAPTER IV UNIVERSAL ALGEBRA

4.1D. Verify that TB in Definition 4.1.5(c) and "5 in Definition 4.1.6(c) really are categories. 4.1E. Let T be a closure operator on a poset (B, ^ ) . Show that the corresponding Eilenberg-Moore category is the induced poset of closed elements, i.e. of fixed points of T. 4.1F. Let K be a prevariety. Show that each |f-morphism / : X -» X' yields a morphism f(J:(XU, e%) ~* (X'U, €%•) in the Eilenberg-Moore category of the monad (ClRgU, SlR^eU, TJ) of K. 4.1G. Verify the claim of Example 4.1.8. 4.2. Monadic Adjunctions Starting with a monad (T, \i, TJ) on a category B, the Eilenberg-Moore Theorem 4.1.7 builds the adjunction (5, R, TJ, ê) that recovers the monad as (SR, SëR, TJ). Conversely, one may start with an adjunction (F : B -» A, G : A -* B, TJ, e) and consider the monad (T, p, e) — (FG, FeG, TJ) on the base category B. The following question then arises: How is the original category A related to the Eilenberg-Moore category **£? To address the question, one may summarize the data by the outer rectangle in the following diagram: B —$—* »B —$-+ B (4.2.1) B

-» A

G

-* B

The bottom row records the original adjunction (F, G, TJ, e), while the top row records the Eilenberg-Moore adjunction (S, R, TJ, 2) of the monad (4.2.2)

(FG,FeG,T)) =

(7\/LI,TJ)

= (SR,SeR,

TJ).

The relationship _between A and ^B is then expressed by a uniquely specified functor G : A -> ^B. Proposition 4.2.1. Given an adjunction (F: B -» A,G: A -» B, TJ, e) yielding the monad (4.2^2) and Eilenberg-Moore adjunction (5, R, TJ, e), there is a unique junctor G : A -» Mfl, called the Eilenberg-Moore comparison of the adjunction (F, G, TJ, e), such that FG = S and GR = G.

MONADS

353

Proof. The Eilenberg-Moore comparison is defined (by analogy with Exercise 4.IF) to have morphism part G : ( J t : Y - Y ' ) ~ (kG:(YG,e°)

(4.2.3)

^(Y'G,EG.)).

For Y in A0, verification of the associative law (FEG)YGEY = EYFGBY in the FG-algebra (YG, EY) proceeds by applying the functor G to the naturality EYGFEY= EGFEY °f e'.GF -* \A at the A -morphism eY : YGF -* Y. [Note that (4.1.1) is an instance of this verification, with Y = XF.] The unit law riYCeY = 1YG in the FG-algebra (YG, eY ) is the component at Y of the identity (Gr¡Y.eG) = 1 of Proposition III 3.1.7. Thus YG lies in " 5 0 . For k : Y -» Y' in Ax, application of the functor G to the naturality eYk = kGFsr of E : GF -» \A at k shows that fcG lies in ,1BV Verification of the functoriality of G : A -» ^5 is straightforward (Exercise 4.2A). The equation GÄ = G is immediate from the definitions of G and R. On the other hand, for a B-morphism f:X'-*X, substituting fG.X'G^XG for * : Y - > Y ' in (4.2.3) yields (4.1.5) by virtue of (4.22), so that FG = S also holds. To confirm the uniqueness of G, consider a functor X : ^4 -» M5 with FK = S and KR = G. By the latter equation, the morphism part of K takes the form K:(k:Y^Y')

~(kG:(YG,a)

-> (Y'G.o')),

in which a and a7 are respective ¿¿-algebra actions on YG and Y'G, connected by a T-homomorphism kG. For Y in A0, it must thus be shown that the action « of YK coincides with sY. First note that the naturality of e : RS -> 1M# at e* : YKRS = YGS = YGFK -> YK yields s

YKRSeY

=

e

r

£

yjC-

By Proposition III 3.1.7 and the definition of the counit B in the EilenbergMoore adjunction, one then has (a : (YGT, fiYG) -* (YG, a)) = E(YG a) = ~BYK — (VYGeY>

quired.

e

YK

=

r

lYKREY

e

YK ~

r

¡YKReYKRSeY

=

e

Y

=

e

Y>

aS

re

"

D

Definition 4.2.2. An adjunction (F : B -» A, G : A -* B, r¡, s), and the right adjoint G : A -» B in such an_ adjunction, are said to be monadic if the Eilenberg-Moore comparison G : A ->FeGB is an isomorphism of categories. The category A is said to be monadic over the base category B if there is a monadic right adjoint G : A -» B. D Example 4.2.3. As noted in Section III 3.4, the forgetful functor G : Top-> Set is right adjoint to the discrete topology functor D : Set-» Top.

354

CHAPTER rV UNIVERSAL ALGEBRA

However, this adjunction yields the trivial monad (1««, 1 : Is« ~* !§«> 1 : lcj,-» lfc.) on the base category Set. The Eilenberg-Moore category of this monadis just Set, and the Eilenberg-Moore comparison is just the forgetful functor G : Top -» Set, which is not an isomorphism. Thus the right adjoint G :Top-> Set is not monadic. □ Theorem 4.2.7 below shows that, for any theory map M : W -» V, the functor Mr : V-* ¡V (known by Theorem 3.4.4 to be a right adjoint) is monadic. The proof of Theorem 4.2.7 relies on a certain characterization (Proposition 4.2.6) of those T-algebras that are /¿-algebras for a monad (T, ¿1,17). The characterization uses a special kind of coequalizer. Prosition 4.2.4. If the diagram

A2 (4.2.4)

T,

* Ai

? fx

* A2

a ■+A,

Q

in a category D commutes, then q : A2 -* Q is the coequalizer of the parallel pair (/1./2).

Proofs Given a Z>-morphism f:A2-*B with fxf = f2f, define f=s0f. Then qf - qsQf = sxf2f = s , / , / = /. Moreover, if / ' :Q -» B satisfies qf' — f, one has / ' = s0qf' = s0f = /. D Corollary 4.2.5. If (4.2.4) commutes in D, then qE is the coequalizer of the parallel pair (fxE, f2E) for each functor E: D -» C with domain D. Proof. Under the functor E, the commuting diagram (4.2.4) in D translates to a commuting diagram in E. D A coequalizer q : A2 -* Q of a parallel pair (/, : Ax -* A2, f2:Ax~* A2) is described as split by morphisms s0 : Q -* A2 and sr : A2 -* Ax if the diagram (4.2.4) commutes. For example, if a is an equivalence relation on a set A, then the coequalizer nat a : A -> A" in Set of the parallel pair (TTU TT2) of

MONADS

355

projections, with TT, : A2 -+ A; («j, a2) -* a¡ for 1 ^ i á 2, is split by a section s : A" -* A of nat a and the diagonal embedding A : A -» /I 2 . Proposition 4.2.6. Lei (r, /x, 17) fee a monad on a base category B. Then a T-algebra (X, a) is a Li-algebra iff a : XT -» Xis a coequalizer of the parallel pair ( fix, aT) split byT)x:X^XT and T)XT:XT -* XT2. Proof. Consider the instance l

XT

XT

VxT

* XT2

y-x

■* XF

aT

(4.2.5) Vx

•* XT

-» X

1* of (4.2.4). Suppose, for the T-algebra (X, a), that (4.2.5) commutes. Then the associative law is satisfied by the commuting of the right-hand square, while the unit law is satisfied by the commuting of the bottom rectangle. Conversely, suppose that (X, a) is a /¿-algebra. The right-hand square and bottom rectangle of (4.2.5) commute respectively by the associative law and unit law in the /x-algebra (X,a). The left-hand square commutes by the naturality of TJ : 1B -* T at the action a : XT -» X, while the top rectangle commutes by the component at X of the right unit law of the monad (T, LL, TJ).

D

Theorem 4.2.7. Let M:W M' : V-* W is monadic.

V be a theory map. Then the right adjoint

Proof. By Theorem 3.4.4, the theory map M:W -» V yields an adjunction (M', M\ 17, e). For the monad (7\ p, TJ) = (M'Mr, M'eMr, TJ) of this adjunction, Proposition 4.2.1 yields the Eilenberg-Moore comparison functor W : V-* " W, Y ~» (YMr, eYMr). The proof of Theorem 4.2.7 thus consists of the provision of a two-sided inverse functor H : ^W^ F to IF. To this end, it is most convenient to recognize F a s a variety of algebras of the type T : il -» M of V [cf. (3.3.1)], according to Theorem 3.3.3. Let (X, a) be a /n-algebra, and let on X. Note that the K-algebra structure on XM1 yields the operation w : (XM')aT -* XM'. (Here and elsewhere in this proof, explicit mention of the forgetful functors U : £-> Set and U : W_-+ Set is often suppressed.) Since

356

CHAPTER IV UNIVERSAL ALGEBRA

the right adjoint M' is continuous, and commutes with the forgetful functors by Corollary 3.4.3 [and Exercise 3.4F(b)], one thus obtains XT, and similarly
2

(AT ) (4.2.6)


(aT)'

?(AT)

»T

a"

-» XmT (Oi

1

' AT 2

aT

i XT

-* X

In each of the horizontal rows, the right-hand arrow is the coequalizer of the parallel pair to its left, according to Corollary 4.2.5. Indeed, starting with the split coequalizer (4.2.5) in W, the bottom row is obtained by applying the functor U : W~* Set. The top row is obtained by applying the composite functor (4.2.7)

(an) =

{W-±W'^W^S$,

with / an antichain poset category or set of cardinality >XTTaTa = ù)XTT(aM')a = (aM')XTa. Since the top row of (4.2.6) is a coequalizer in Set, it follows that there is a unique function XTa = a^to. This function provides the operation of
an(u, n)x = (u, n)XTa = (u, n)XTa = a"(v, n)x,

since the K-algebra AT or XM1 satisfies u = v. Application of the functor («) = ((u, «)T) of (4.2.7) to the bottom rectangle of (4.2.5) yields r¡xa" = I*», so that a" : (AT)" -» X" surjects. Equation (4.2.8) thus implies (u, n)x = (v, n)x : The fl-algebra (X, fl) satisfies u = v. In other words, (X, a)H is an object of V. Moreover, (X, (fitíW recovers (X, a), since e(X a)H is a coequalizer of fix and aT.

MONADS

357

Finally, consider a /x-homomorphism h : (X, a) -* (Y, b), so that ah = hTb. For an operator Y = a^h^Wy. Sincefl
(b) (c)

A variety F of algebras of type T : il -* N; The category F of models of the algebraic theory V; The Eilenberg-Moore category F of the monad of the adjunction (SIRy, U,T),E).

EXERCISES 4.2A. Complete the_ verification of the functoriality of the Eilenberg-Moore comparison G : A -»^B in the proof of Proposition 4.2.1. 4.2B. Consider a pair (F : B -» A, G : A -* B, v, e) and (S : B -* C, R : C -> B, 17, ê) of adjunctions with common unit 17. If there is a functor K : A -* C with FK = S and KR = G, show that the two monads (FG, FeG, 17) and (SR, Se R, TJ) on B agree. 4.2C. Show that a Galois connection (S : B -» A, R : A -* B, 17, e) is not monadic if A contains elements that are not closed. 4.2D. (a) If q : A2 -* Q is a coequalizer of the parallel pair (/, : Ax -* A 2»f2:Ai -*A2^> show that it is also the coequalizer of the parallel pair (f2,fi)(b) Give an example of a morphism a : A2 -* Q that is a split coequalizer of a parallel pair (ft : Ax -* A2, f2:Ax~* A2), but not of the parallel pair (/ 2 ,/,).

358

CHAPTER IV UNIVERSAL ALGEBRA

4.2E. Let 1 and 2 respectively denote the posets with Hasse diagrams 0 and 0 -* 1, considered as small categories. Let C be a small category with terminal object T. Show that the unique functor C -* 1 is the coequalizer in Cat of the parallel pair (C 2 -* C; F -* OF, C2 -* C; F -> IF) of functors, split by the functors 1 -» C with object part 0 -» T and C -* C2 with object part x -» ((0 -» 1) -» (x -» T)). 4.2F. Complete the verifications required for the proof of Theorem 4.2.7. 4.2G. Show that the Eilenberg-Moore comparison of the adjunction (ilRp U, J\, e) determined by a prevariety K is the inclusion of K in the variety HK. 4.2H. Let (T, p., TJ) be a monad on the base category Set such that ** Set is a variety of algebras. Characterize those properties of the monad (T, /x, TJ) that correspond to entropicity of the algebras in the variety MSet. (Hint: Cf. Exercise 2.4D.) 4.21. Let T : il -* N be a finitary type. Let {T, p, r¡) be the monad of the adjunction (ÍÍ : Set-» T, U : T-» Set , TJ, e). Show that the categories T, r Set. and ** Set coincide. 4.2J. Let (7\ ¡j., rj) be a monad on the base category Set. Show that ** Set is a variety of algebras of finitary type iff VF e Set,,, V y e Y T , 3 ( 9 :{*„...,*„} - * y ) G § e t r 3 * e {xt

xn}T. x
4.2K. A subcategory A of a category B is called abstract (cf. Definition 2.1.1) if each B-object isomorphic to an .¿-object in B is itself an object of A. (a) If the embedding of an abstract subcategory A in B is a reflection (cf. Exercise HI 3.2.E), show that the embedding is monadic. (b) For an algebraic theory V, show that V is monadic over V. 4.2L. Let R : A -* B be monadic, and let F : J -* A be a functor with small domain. Show that R creates the limit of F (cf. Exercise III 3.4B).

INDEX A page number repeated within an entry denotes that the keyword on that page has been used in different contexts. ¿„,80 ¿.,81 abelian, 70 absolute value, 156 absolutely free, 291 absorption law, 216 abstract, 304, 358 action, 29, 349 action morphism, 38 Adconj, 342 addition formulae (of trigonometry), 122 adjoint, 245, 250 adjunction, 245. 250 affine equation, 137 affine group, 238 Alexandrov topology, 270 Alg, 188 AUj.202 algebra, 158, 284 algebra homomorphism, 159 algebraic (algebra), 187 algebraic (element) 187 algebraic multiplicity, 185 algebraic number, 187 algebraic theory, 324, 336 algebraic theory of a prevariety, 336 algebra in category, 319 algebra morphism in category, 320 algebra of power series, 241 almost zero, 118 alphabet, 20, 20 alternate, 150 alternating functional, 150 alternating group, 80, 81 alternating group action, 80, 81 alternative law, 94 An, 132 annihilator, 132, 261 antichain, 24, 203 antireflexive relation, 73, 73

antisymmetric relation, 10 antitone map, 201 arg, 292 argument, 292 argument map, 292 arithmetic mean, 52 arity, 284 arrow, 191 ascending chain condition, 196 associate, 168 associative, 17,133, 192, 348, 350 atom, 260 atomic, 260 Atp, 91 augment, 287 augmentation, 224 augmented matrix, 139 Aut, 66 automaton, 34 automorphism, 66, 72 automorphism group, 66, 72 autotopy, 91 axiom rank, 317 BAlg, 257 Banach Decomposition Theorem, 222 base category (of a monad), 348 base rank, 317 basic operator, 293 basic reduct, 287 basis, 145 batteries, 283 Bell numbers, 78 bicomplete category, 243 bihomomorphism, 321 biject, 6 bijective function, 6 bilinear function, 150 bilinear functional, 150

359

360

INDEX

bilinear pairing, 150 binary alphabet, 20 binary operation, 282 binary relation, 11, 72 biproduct, 115, 116,131 biregular representation, 44 Birkhoff's Variety Theorem, 314 black boxes, 283 block, 318 block code, 104 Bol identity, 97, 98 Bol loop, 97, 99 Boolean algebra, 257, 267 Boolean algebra homomorphism, 257 Boolean ring, 256 bounded, 221 bounded above, 221 bounded below, 221 bounded chain, 267 bounded lattice, 221 bounded lattice homomorphism, 221 BRing, BRing*", 257 Burnside's Lemma, 75 burst error, 109 C C 55 C„-, 244 cancellative, 104 Cantor, 196 Cantor-Schröder-Bernstein Theorem, 222 Card*", 254 cardinality (of a multiset), 50 Cartesian product, 7 Cat, 201 category, 191, 197 category of constant functors, 226, 226, 239, 240 category of elements, 330 category of partial functions, 198 category of quasigroups, 197 category of relations, 199 category of small categories, 201 Cauchy, 75 Cauchy's Theorem, 83 Cayley-Hamilton Theorem, 182 Cayley's Theorem, 43, 330 center, 68 centralize mutually, 57 centroid, 323 Cg,288 chain, 16 characteristic function, 13 characteristic (of an algebra), 162

characteristic polynomial, 183 Chinese Remainder Theorem, 181 circle group, 137 circuit breakers, 283 class equation, 70 class of r-algebras, 304 clonally equivalent, 295 clone, 293, 293 clone operation, 293 closed element, 261 closed interval, 194 closed operator domain, 293 closed permutation group, 75 closed set of operations (of an algebra), 293 closed (set of operations on a set), 293 closed subset, 270 closure, 17, 41, 261, 292, 293, 293 closure operator, 348 coalgebra, 342 (»complete category, 243 «►continuous functor, 266 Cod, 4 code, 21, 104 coding homomorphism, 21 codomain (of a function), 4 codomain (of a morphism), 191 coefficient, 147 coefficient matrix, 139 «»equalizer, 228 cofinite, 217 cofmite topology, 270 Coker, 112 cokernel, 112, 131, 229 colimit, 240 colimit morphism, 240 column matrix, 133 column (of matrix), 119 column operation, 141 column rank, 143 column reduced, 143 combinatorial (morphism), 341 combinatorial multiplication group, 55 combinatorial right multiplication group, 101 comma category (of functor over object), 248 comma category (of object over functor), 247 comma code, 21 commutative, 24, 123, 236 commute, 205 complement, 256 complete category, 243 complete Heyting algebra, 269 complete lattice, 221 completely distributive lattice, 269 completely reduced form, 46

INDEX Completeness Theorem of Equational Logic, 316 complex conjugate, 156 complexification, 280 complex multiplication, 17 complex numbers, 156 component (of a matrix), 119 component (of a natural transformation), 210 componentwise, 19, 37, 57, 114, 219, 301 composite functor, 201 composition, 191, 348 compound action, 34 compound event, 34 concepts, 264 concrete category, 202 congruence, 11, 19, 32, 60, 101, 216, 287, 307 congruence generated, 288 congruence modulo, 11 congruence poset, 288 conjugacy class, 66 conjugate, 156, 157 conjugate module, 150 conjugation, 66, 156 constant, 159, 282 constant functor, 225, 226, 226, 230, 231, 239, 248 continued fractions, 303 continuous function, 270 continuous functor, 266 contravariant functor, 201 contravariant power set functor, 201 convex sets, 294, 298, 308 convolution, 196, 227 cooperation, 343 coproduct, 115, 116, 117, 131, 215, 220, 220 core, 67 correct (error), 105 corresponding homogeneous equation, 137 coset, 42 coset leader, 106 counit (of adjunction), 247, 250 countable alternating, 81 countable cycle, 55 countable dihedral, 56 countable permutation action, 78 countable permutation group, 78 countable set, 198 covariant functor, 200 covariant power set functor, 204 cover, 194 covering pair, 194 create (colimit), 272 create (limit), 270 CRing, 225

cubic element, 187 Currying, 12, 250, 331 cycle, 55 cycle decomposition, 58 cycle notation, 57 cycle type, 58 cyclic action, 55 cyclic group, 55 cyclic monoid, 23 D„, A., 56 decision procedure, 46 decoding, 105, 169 deduction rules, 316 deg, 159 degree (of an algebraic element), 187 degree (of a permutation group), 72 degree (of a polynomial), 159 delta function, 49, 118, 135 delta polynomial, 161, 299, 300 de Morgan Law, 125 dense functor, 254 dependent, 145 derived, 292 derived operation, 293 derived operator, 293 det, 152 determinant, 93, 152 determinant of an endomorphism, 153 diagonal, 11, 298 diagonal functor, 225, 230, 239 diagonal line, 89 diagonal matrix, 121, 126 diagonal transformation, 239 diagonalization argument, 196 diagram, 205 dihedral action, 56 dihedral group, 56, 186, 237 dim, 153 dimension, 105, 153, 156, 238, 238, 290 directed edge, 73 directed graph, 73, 194 directed poset, 240 direct limit, 240 direct power, 8 direct product, 7, 19, 37, 56, 114, 216 direct sum, 115, 116, 117 direct summand, 116 discrete category, 203 discrete logarithm, 189 discrete poset, 203 discrete topology, 269 discrete topology functor, 270

361

362

INDEX

disjoint union, 8, 9, 31, 50 distance, 105 distribution, 135 distributive lattice, 256 distributive law, 122, 122, 256, 256 diversity relation, 73, 73 dividend, 168, 169 division, 60 Division Algorithm, 168 division by zero, 165 division ring, 127 divisor, 168, 169 Dod, 198 Dom, 4 domain, 128 domain (of a function), 4 domain (of a morphism), 191 domain of definition, 198 dominate, 273 down-set, 200 dual, 134, 134,150, 193 duality, 6,106,134 dual poset, 194 dynamical system, 22, 181 E 2 , 297 ED, 168 edge, 73, 73 effective, 72 eigenspace, 184 eigenvalue, 184 eigenvector, 184 Eilenberg-Moore adjunction, 350 Eilenberg-Moore category, 350 Eilenberg-Moore Theorem, 350 electromagnetic waves, 100 elementary action, 34 elementary column operation, 140, 141, 141, 141 elementary divisor, 178, 184 elementary event, 34 elementary matrix, 119 elementary reduction, 46 elementary symmetric function, 266 element (of a multiset), 50 empty word, 20 encoding, 105 End, 113 endomorphism, 113 English alphabet, 20 enrichment, 287 entropie, 63, 318 entry (of matrix), 119

epigraph, 292 epimorphism, 5, 206 Eq, 315 equalizer, 231 equational theory, 315 equipollent, 6 equivalence, 254 equivalence class, 10 equivalence relation, 10 equivalent (algebras), 295, 295 equivalent categories, 254 equivalent module homomorphisms, 175 equivalent systems, 139 Erlanger Programm, 298 error, 105, 169 error map, 105, 169 Euclidean Algorithm, 178 Euclidean domain, 168 Euclidean motion, 297 Euclidean plane, 297 Euclidean topology, 270 evaluation map, 137 evaluation natural transformation, 227 even permutation, 80 evolution operator, 22 exact, 112 exact at, 112 exact sequence, 112 existence, 138 exponent, 49 exponential function, 122, 200, 300 exponentiation, 12 extending independent set to a basis, 148 extension of a function, 6 extension of an abelian group, 117 extension relation (between partial functions), 199 Extension Theorem, 334, 345 extent of a concept, 264 external direct sum, 115, 116, 117 factorization, 175 faithful (functor), 201 faithful (G-set), 72 faithful (module), 129 falling factorial, 74 fictitious, 301 field, 127 field of algebraic numbers, 188 field of fractions, 165 finitary algebraic theory, 336 finitary type, 284 finitely generated module, 174

INDEX First Bol Identity, 98 First Isomorphism Theorem, 12, 20, 32, 62, 104, 128, 131, 288 First Isomorphism Theorem for Abelian Groups, 112 First Moufang Identity, 93 Fix, 31 fixed point, 31 fixed-point free, 31 fixed-point functor, 262 flag, 16 Fid, 209 flexible identity, 92 forgetful functor, 202 form, 150 formal power series, 241 fractions, 164ff fractions (continued), 303 free algebras, 308, 342 free commutative monoid, 49 free commutative semigroup, 50 free group, 45, 308 free group functor, 253 free group generated by a set, 48 free G-set over ¿/-set, 63 free module, 130, 146, 146 free monogenic module, 130 free monoid, 20 free monoid functor, 251 free A/-set, 33 free presentation, 311 free resolution (of a module), 177 free semigroup, 20 Freyd Adjoint Functor Theorem, 274 Frobenius, 75 Frobenius automorphism, 189 Frobenius homomorphism, 189 full (functor), 201 full (subcategory), 201 fully invariant congruence, 315 functional, 150 functionally complete, 299 functor, 200 functor category, 211 Fundamental Theorem of Arithmetic, 51, 168 Fundamental Theorem of Galois Theory, 264 GA„, 238 Galois connection, 260 Galois correspondence, 261 Galois field, 189 Galois group, 262 Galois Theory, 262

Gaussian elimination, 141 Gauss' Lemma, 172 general affine group, 238 generalized function, 135 general linear group, 156 general solution, 138 generated submonoid, 20 generator, 48, 80, 230 geometric multiplicity, 185 geometric realization, 342 GL B , 156, 290 gib, 24 good definition, 12 Gp, 201 graph, 73, 194 graph of a (partial) function, 199 greatest lower bound, 24 Grimeisen operator, 203 group, 41 group conjugacy class, 66 group in a category, 236 group kernel, 66 group presentation, 229 group with (given) presentation, 229 group of units, 45 groupoid, 284

H, 304 Hadamard product, 124 Hamiltonian group, 70 Hamming distance, 105 Hamming graph, 74 Hamming weight, 105 Hasse diagram, 194 head function, 191 Heaviside function, 137 Heyting algebra, 257 Hoi, 239 holomorph, 239 Horn, 112 homogeneous linear equation, 137 homogeneous space, 42 homomorphism, 18, 31, 60, 101, 101, 123, 123, 130, 133, 159, 216, 221, 257,285 homomorphism of T-algebras, 349 homothety, 130 homotopy, 86 homotopy category of quasigroups, 198 Hopf algebra, 344 horizontal, 89 hypercube, 74

363

364

INDEX

/, 304 id, 5 Id, 126 ideal, 126 ideal generated, 126 idempotent, 23, 24, 63 identity, 313 identity element, 17, 87, 101 identity function (of a category), 191 identity function (on a set), 5 identity functor, 201 identity matrix, 120 identity transformation, 211 Im, 112 image, 261 imaginary part, 156 impoverishment, 287 improper, 31, 32, 61, 126, 287 improper ideal, 126 incidence matrix, 126 inclusion functor, 201 inconsistent, 138 indecomposable, 31 independent, 145 indeterminate, 159 index, 23 indiscrete topology, 269 indiscrete topology functor, 270 induced G-set, 63 Induction Basis, 15 Induction Hypothesis, 15 Induction Step, 15 inductive limit, 240 inductive poset, 16 inertial subgroup, 42 infix notation, 282 inhomogeneous linear equation, 137 inhomogeneous term, 137 initial object, 207 inject, 5 injective function, 5 injective module, 149 Inn, 66 inner automorphism, 66 inner automorphism group, 66 insertion, 8, 115, 117, 215, 220, 240 integer partition, 50, 253 integral domain, 128 intent of a concept, 264 internal direct sum, 115, 116, 117 invariant, 31, 297 invariant factor, 178, 183 inverse functor, 201 inverse image, 261

inverse limit, 240 inverse (of a function), 6 inverse (of an isomorphism), 207 inverse property, 91 inverse transformation, 211 inversion, 42, 236 invertible element, 45 invertible function, 5, 6, 6 invertible morphism, 206, 206, 206 Irr, 168 irreducible (element of commutative ring), 168 irreducible (A/-set), 31 Iso, 44 isomorphic, 6, 18, 31, 201, 207 isomorphism, 6, 18, 31, 201, 206, 224 isomorphism of subobjects, 224 isotope, 87 isotopic, 87 isotopy, 86 isotropy subgroup, 42 item, 264 Johnson graph, 74 join, 216 join distributive law, 256 join semilattice, 215 Jordan canonical form, 184 ker, 12 Ker, 66 kernel, 12, 66, 262 knot group, 342 Krasner-Wielandt Theorem, 75 Kronecker delta function, 49 Kuratowski, 16 ^,312 LA, 297 Lagrange interpolant, 161, 299 Lagrange Interpolation, 161 Lagrange's Theorem, 44 Laplacian, 100 large category, 197 Lat, 216 Latin square, 53 lattice, 215 lattice homomorphism, 216 leading coefficient, 159 least upper bound, 215 left adjoint, 245, 250 left alternative law, 94 Left Bol Identity, 98 left Bol loop, 99

INDEX left cancellative, 104 left distributive law, 122 left division, 60 left inverse, 6, 206 left inverse property, 91 left invertible function, 6 left invertible morphism, 206 left module, 133 left module homomorphism, 133 Left Moufang Identity, 93 left Af-action, 29 left Af-set, 29 left multiplication, 18, 124 left regular representation, 43, 133 left unit law, 347, 348 length (function of Euclidean domain), 168 length (of a channel), 104 length (of a partition), 50 length (of a word), 20 lifting a homomorphism, 149, 308 lim, 240, 240 limit, 240 limit morphism, 240 linear algebra, 11 Iff, 256 linear code, 106 linear combination, 145, 145 linear dynamical system, 181 linear equation, 137, 137 linear functional, 150 linear ordering), 15 linear representation, 237 local duality, 106 localization, 165 locally finite (poset), 194 locally small, 197 local ring, 166 logarithmetic, 312 loop, 87 loop transversal, 102 Lp, 202 ¡üb, 215 M, M, 340 M1, 342 AT, 340 macroscopic, 40 Ai-action, 29 magma, 284 majority vote, 105 matrix, 9, 119 matrix algebra, 158 matrix product, 124 Max, 127

maximal, 16 Maxwell's equations, 95 A/-congruence, 32 meet, 216 meet distributive law, 256 meet semilattice, 215 M-homomorphism, 31 ju-algebra, 350 /¿-homomorphism, 350 microscopic, 40 minimal polynomial, 183, 187 Minkowski spacetime, 95 Af-isomorphism, 31 mixed associative law, 133, 323 Mit, 55 Möbius function, 196 Mod, 202, 202 Mod* ", 212, 256 model of an algebraic theory, 336 module, 129, 237 modulus, 156 Mon, 224 Mon, 202 monad, 347, 348 monadic adjunction, 353 monadic category over base category, 353 monadic (right adjoint) functor, 353 monad of a prevariety, 349 monic (polynomial), 159 monoid, 17 monoid congruence, 19 monoid homomorphism, 18 monoid in a category, 236 monoid isomorphism, 18 monid ring functor, 345 monomial, 159 monomorphism, 5, 206 monotone (increasing), 200 (Mon; Set), 39 Mor, 19T morphism, 191, 236 morphism part (of functor), 200 morphism set, 191 Morse alphabet, 20 Morse Code, 22 Moufang element, 92 Moufang identity, 92, 93, 93 Moufang loop, 92 A/-subset, 31 multiplication, 17, 51, 101, 124, 124, 236 multiplication group, 55 multiplication table, 72, 72 multiplicity of occurrence, 49

365

366

INDEX

multiplying out, 171 multiset, 49 multiset form (of a partition), 50 nat, 11 natural isomorphism, 210 natural projection, 11 natural transformation, 209 nerve, 195 net, 88 Net, 88 nofix notation, 282 non-associative, 52 non-trivial path, 194 non-unital module, 129 non-unital ring, 122 non-unital ring homomorphism, 123 non-unital subring, 123 norm, 156, 187 normal form, 46 normalized transversal, 102 normalizer, 67 normal subquasigroup, 66 nullary, 282 0 2 , 297 H, 292 n-algebra, 284 Obj, 191 object, 191 object over, 223 object part (of functor), 200 object representing a functor, 328 object set, 191 object under, 224 octahedral graph, 74 odd permutation, 80 fi-word, 291 one-point compactification, 203 onto, 5 open half-space, 270 open interval, 194 open subset, 269 operation on a set, 282 operation symbol, 284 operator, 284 operator domain, 284 opposite, 29, 133, 193 orbit, 23, 31, 41 orbital, 65 order, 10 ordered set, 10 order (of group element), 59

order-preserving, 200 order-reversing, 201 outer automorphism, 66 overdetermined, 138

p, 304 p-adic integers, 243 parallel pair, 228 parametrization, 12 parity check matrix, 120 partial function, 198, 225 partial homomorphism, 107, 157 partial ordering), 10 particular solution, 138 partition, 50, 50, 253 part (of a partition), 50 path, 194 path category, 194 period, 23 permutable, 25, 57 permutation, 58, 78, 140 permutation character, 76 [a] permutation group on a set, 72 [the] permutation group on a set, 58, 78 permutation matrix, 121 permutation polynomial, 163 Petersen graph, 74 Pfn, 198 PGL„, 290 p-group, 82 n„,285 PID, 169 Pigeonhole Principle, 16 pivot, 141 plain, 287 poclass, 198 Poincaré series, 76 point, 191 pointwise, 218 Pointwise Limit Theorem, 332 polarity, 261 Polish notation, 282 polynomial, 159 polynomial function, 160 polynomially complete, 299 polynomially equivalent, 295 polynomial operation, 295 polynomial ring, 159 Pos, 203 poset, 10 postfix notation, 282 power series, 241

INDEX power set, 10 prefix notation, 282 pre-ordering, 41 presentation (of groups), 229 preserve (colimit), 266 preserve (limit), 266 preserve (relation), 297 preserve (set of relations), 302 presheaf, 327 prevariety, 307 prevariety generated by an algebra, 312 primary decomposition, 178 primary module, 178 prime element (of integral domain), 169 prime field, 188 prime ideal, 127 primitive element, 189 primitive M-set, 32 primitive polynomial, 171 principal ideal, 126 principal ideal domain, 169 principal isotopy, 87 Principle of Complete Induction, IS Principle of Local Duality, 106, 107 Principle of Mathematical Induction, IS product, 7, 7, 9, 31, 114, 124, 124, 214, : 219, 287 product category, 331 product form (of a partition), 50 projection, 7, 11, 214, 218, 240, 285 projection functor, 330 projection operator, 293 projection representation, 285 projective general linear group, 290 projective limit, 240 projective module, 149 projective special linear group, 290 projectivity, 149, 308 Prop, 127 proper class, 197 proper ideal, 126 proper suffix, 21 property, 264 pseudocomplement, 267 PSL,,, 290 pullback, 232 pushout, 232 Q, 197 Qtp, 198 quadratic element, 187 quandle, 342 quasigroup, 52, 60

367

quasigroup congruence, 60 quasigroup conjugacy class, 66 quasigroup homomorphism, 60 quasigroup homotopy, 86 quasigroup isotopy, 86 quasi-ordering, 41 quaternary, 72, 282 quaternions, 157 quotient, 11, 168, 169, 288 quotient module, 131 fl*,304 rank, 143, 153, 178 rational canonical form, 183 rational (functions), 165 real part, 156 rectangular band, 339 reduced system, 139 reduct, 74, 287, 293, 395 reduction, 139 reduction functor, 345 reflection, 259 reflexive relation, 10 regular (subgroup of a permutation group), 83 reinforced concrete (category), 203 Rel, 199 relation, 10, 72 relation (of an algebra), 311 relational structure, 72 relation imposed, 229 relation monoid, 25 relation product, 25, 199 remainder, 168, 169 remote control, 34 repetition code, 105 replica functor, 304 replication, 304, 342 representable functor, 328 representation, 29, 129, 237 representation of T, 285 reseating, 140 residue field, 166 restriction (functor), 287 restriction (of a function), 6 restriction (of a group action), 63 retraction, 5, 206 reverse Polish notation, 282 Riemann integral, 135 right adjoint, 245, 250 right alternative law, 94 Right Bol Identity, 97 right Bol loop, 97 right coset, 42

368

INDEX

right distributive law, 122 right division, 60, 101 right inverse, 5, 206 right inverse property, 91 right invertible, 206 right invertible function, 5 right loop, 101 right loop homomorphism, 101 right Ai-action, 29 right module, 129 right module homomorphism, 130 Right Moufang Identity, 93 right A/-set, 29 right multiplication, 18, 124 right quasigroup, 101 right quasigroup homomorphism, 101 right regular representation, 29, 43, 124, 133, 159 right subloop, 101 right subquasigroup, 101 right transversal, 42 right unit law, 347, 348 rigid graph, 74 rigid motion, 298 ring, 122 Ring. 202 ring homomorphism, 123, 123 ring-with-a-one, 122 ring-without-one, 122 RLp, 202 RNÄ alphabet, 20 Rng, 202 root, 161 row matrix, 130 row (of matrix), 119 row operation, 179 RQ, 202 s, 304 S„, S„, 78, 78 ¡y, 329 Sarrus' Rule, 155 satisfy an identity, 313 satisfy the solution set condition, 274 Sb, 82, 132, 286 scalar, 130 scalar matrix, 125 scalar multiplication, 130 Second Bol Identity, 97 Second Isomorphism Theorem, 289 Second Moufang Identity, 93 section, 6, 206 Segre characteristic, 51, 253

semigroup, 17 semigroup homomorphism, 18 semigroup in a category, 236 semilattice, 24 separable, 188 sequence of groups, 237 Set, 12, 197 set of elements of a multiset, 51 sgn, 81 Sgp, 202 shear, 141 short exact sequence, 117 Sierpiiiski space, 270 signature, 284 sign (of a permutation), 81 similar endomorphisms, 181 similarity, 38,181 simple, 32, 61, 126, 287 simple directed graph, 73 simple ring, 126 simple undirected graph, 73 simplex, 174, 195, 308 Simpson's Rule, 163 sink, 126, 286 sink generated, 288 sink poset, 287 size (of a multiset), 50 Sk,287 skewfield, 127 SI, 202 SL„, 156 slice category, 223, 224, 248 (small) bicomplete, 243 small category, 191 small cocomplete, 243 small complete, 243 solution set, 137, 274 solution set condition, 274 solving, 137 source code, 104 space (of a representation), 237 span, 146 spans, 145 Spec, 127, 257 special affine group, 238 special linear group, 156 spectrum, 127 split coequalizer, 354 split exact sequence, 117 split extension, 117, 237 splitting field, 188 square (matrix), 119 stability congruence, 68

stabilizer, 42 stabilizer functor, 262 state, 22 Steiner triple system, 318 stopwatch, 35 strictly simple, 287 Structure Theorem for G-sets, 43 STS, 318 subalgebra, 286 subalgebra generated, 288 subalgebra poset, 286 subcategory, 201 subgroup, 42, 64 subgroup generated, 48 sublattice, 216 submodule, 130 submonoid, 19 subobject, 224 subquasigroup, 60 subring, 123, 123 subsemigroup, 19 subspace topology, 270 suffix code, 21 sum form (of a partition), 50 summand, 116 sum (of a partition), 50 sum (of a Segre characteristic), 51 superalgebra, 286 surject, 5 Syl, 82

Sylov's Theorem, 83 Sylov subgroup, 82 symmetric function, 265 symmetric group, 78, 78 symmetric matrix, 121 symmetric relation, 10 symmetry group, 72 syndrome, 108 synonymous, 313 system of linear equations, 139 Í..284 tail function, 191 7"-algebra, 349 T.286 T\ 292 T, 292 T-algebra, 284 T-algebra in category, 319 T-bihomomorphism, 321 T-congruence, 287 T-homomorphism, 285 T-relation, 287, 288

T-word, 291 T-word algebra, 291 Tarski Fixed Point Theorem, 222 tensor product, 278, 320, 321 tensor product algebra, 312 term, 293 terminal object, 207 term operation, 293 ternary, 72, 282 theory map, 340 Third Isomorphism Theorem, 289 Third Moufang Identity, 92 r-homomorphism, 349 threshold gates, 283 Top, 270 topological closure operator, 351 topological group, 272 topological space, 269 topology, 269 topology generated by base, 270 topology spanned by base, 270 topology spanned by subbase, 270 topomorphism, 92 Tor, 180 torsion, 310 torsion element, 180 torsion-free, 354 torsion module, 178, 310 total ordering), 15 total ring of fractions, 167 Tpm, 92 tr, 125 trace (of an algebraic element), 187 trace (of an integer partition), 82 trace (of a matrix), 125 transcendental, 187 transition diagram, 34 transitive, 10, 73 transitive closure, 11 transitive G-set, 41 transitive relation, 10 translation, 123 transpose, 136 transposition, 80 transversal, 42 tree, 323 triality relations, 96 triple, 318, 347, 348 trivial, 30, 31, 32, 61, 126 trivial action, 30 trivial congruence, 287 trivial ideal, 126 trivial path, 194

370

INDEX

truth value, 13 -tuple of elements (of a module), 145 Twoup, 209 type, 284 type derived, 292 type of an algebraic theory, 336 UFD, 169 UMlt, 55 unary, 72, 282 underdetermined, 138 underlying set functor, 202 undirected edge, 73 undirected graph, 73 uniform code, 21 unique factorization domain, 169 uniqueness, 138 unital module, 129 unital ring, 122 unital ring homomorphism, 123 unital subring, 123 unit law, 347, 348, 348, 350 unit (of adjunction), 247, 250 unit (of monad), 348 unit (of monoid), 45 universal geometry, 298 universal multiplication group, 55 universal relation, 11 unknown, 139 unused, 141 upper bound, 16 up-set, 256 used, 142 V, 374 variety, 271, 314 variety of an algebraic theory, 337

vector space, 130 vertex set, 73, 73 vertical, 89 Vierergruppe, 81 Viète's formulas, 266 wall, 286 wall generated, 288 wall poset, 287 weight, 105 well-defined, 12 well-posed, 138 Weyl group, 67 Wl, 287 word, 20 word algebra, 291 wreath product, 40 wreath product action, 40 xor gates, 283 3 (YO). 327 Yoneda embedding, 328 Yoneda functor, 327 Yoneda Lemma, 328 Zariski topology, 272 Zassenhaus' commutative Moufang loop, 93 zero divisor, 129 zero homomorphism, 113 zero matrix, 120 zero object, 207 zero ring, 123 zeta function, 196 Zom's Lemma, 16 Zorn's vector-matrix algebra, 93

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