Relation Algebras

f(a,b)= r x

dx.

Jb Jb

What is /(2a;, x)? If we simply substitute "2E" for "a" and V for "6" without further alterations, we get this statement: rix

f(2x,x) = /

x3dx.

This statement contains both free and bound occurrences of V. To avoid having to distinguish between these two types of occurrences, we follow standard practice in renaming the bound variable before computing the integral.

r

u du = — 4

15

1U

For a similar but more extreme example, see (4.34)-(4.36). The class formation notation is illustrated by the following theorem about the class of sets that are not members of themselves. This theorem implies that if {x ; x $ x} exists, it must be a proper class. Theorem 5. {x : x f. as} is not a set. More formally, X £Aox£x/\xisa

set) ==> A is not a set).

PROOF. Let A be a class. Assume Vx(x € A » a;^a;Aa:isa set). Specialize to A and conclude that A e A o A ^ A A A is a set. It follows from this last statement that if A is a set, then A E A <$ A <£ A, a contradiction. Therefore, A is not a set.

6. Axiom of the Empty Set Glass A is empty if no class belongs to A A is empty <=» Va,(a; $ A) « -Bm{x 6 A). Extensionality precludes the existence of more than one empty set. Theorem 6. If two classes are empty, then they are equal.

42

2. SET THEORY

PROOF. TWO classes A and B have the same elements iff there is no class x that belongs to one of A and B and yet does not belong to the other. If A and B are empty, then there can be no class belonging to one of A and B but not the other, since neither of them has any elements at all. Thus empty classes have the same elements and are equal by the Axiom of Extensionality.

We cannot prove the existence of an empty set from extensionality, so we assume it. The Axiom of the Empty Set is that an empty set exists: The empty set is denoted by the special symbol usually reserved for this purpose, namely "0". The Axiom of the Empty Set can be restated less formally: 0 is a set. The Axiom of the Empty Set implies the existence of a nonempty class, because the empty set must belong to something. 7. Axiom of Complementation Class B is a complement of class A iff the elements of B are the sets that are not elements of A. B is a complement of A <=> Vx(x E B <=> i ^ i A i i s a set). The Axiom of Extensionality implies that

Theorem 7. Every class has at most one complement. The Axiom of Complementation says that every class A does have at least one complement: VA3s(Vj;(a; £ B <=> x is a set A x <£ A)), or, more briefly, VA({X : x £ A} exists). For every class A, let A be the complement of A. (2.1)

A:={x:x(£A}.

Notice that (2.1) is not the same as (1.2). The difference arises from the fact that the universe with respect to which we form the complement of a class in (2.1) is the class of all sets, while in (1.2) the operation of complementation operates relative to a universe consisting exclusively of ordered pairs. The Axioms of the Empty Set and Complementation imply the existence of the class of all sets: Theorem 8. 3vVx(x €V <=> x is a set). On the basis of the previous theorem and the Axiom of Extensionality, define the universal class V by (2.2)

V := {x : x is a set}.

Theorem 9. V ^ i e V f t i i s a set), V = 0, and 0 G V = 0 ^ 0.

S, AXIOM OP INTERSECTION

43

8. Axiom of Intersection Class C is an intersection of classes A and B if C is the class of sets that B). The Axiom are members of both A and B, i.e., ym(x EC-^XEAAXE of Intersection asserts that for all classes A and B there is an intersection of A and B; VAVB3CVX(X EC <&

xeAhxeB).

By the Axiom of Extensionality, there can be at most one such class, so define the intersection A n B by (2.3)

AnB:={x:xeAAxeB}.

therefore, For all classes A and B, A n B exists. Theorem 10. If A is any class, then V = A n A and 0 = A n A. For arbitrary classes A and B, the Axioms of Intersection and Complementation imply the existence and uniqueness of the binary union AU B, the difference A^ B, and the symmetric difference AQ B, where (2.4)

AUB:=AnB,

(2.5)

A~fl:=4nl,

(2.6)

AQB:=(ADB)u(AnB).

Theorem 11. (2.7) (2.8) (2.9) The Axioms of Intersection and Complementation can be used to prove the existence of an empty class, under the assumption that some class exists. They do not, however, entail that the empty class is a set. The Axioms of Intersection and Complementation can be replaced by a single axiom which asserts the existence of the complement of the union of any two classes, that is, A U B exists for any classes A and B: (2.10)

VAVBHGV^S; E C o i i s a set, x <£ A, x <£ B).

The complement and intersection can then be obtained from the complemented union, since ~A~ = AUA,

A n B = AUB U A UB.

The difference axiom 52 (p. 39) is not, by itself, enough to deduce the Axioms of Intersection and Complementation. However, it follows from Tarski-Givant [240, 6.4(iv)] that the existence of intersections and complements follows from the SiSB, SO S2 may replace the Axioms of Intersection and Complementation in our axiomatization of set theory. It is also easy to construct rather more direct proofs of the Axioms of Intersection and Complementation from SI-SB. For the sake of

44

2. SET THEORY

reducing two axioms to one, however, it would be nicer to use (2.10) instead of 9. Calculus of classes The class A is a subclass of the class B, in symbols, A C B, and B is a superclass of A, in symbols B D A, if every element of A is a member of B. A is a subset of B if A is both a set and a subclass of B. A is a proper subclass of B, in symbols, A C B, iff A C B A A ^ B. B is a proper superclass of A, in symbols, B Zi A,ifl B D A and A ^ B. The following theorem gives some common laws involving C and 3 . Much of it can be summarized by saying that U and fl are associative, commutative, idempotent, distribute over each other from both sides, and are dual to each other by way of the involution ~. Its proof uses only the Axioms of Extensionality, Empty Set, Complementation, and Intersection. The consequences of these axioms, especially those exemplified by the statements in the following theorem, constitute "the calculus of classes". Theorem 12. Let A, B, C, and D be any classes. Basic laws: (2.11) (2.12) (2.13) (2.14) (2.15) (2.16)

|CiCV, AdBCA, ACAUB, A~BCA, ACBAACC & ACBCiC, ACCABCC^AUBCC.

Inclusion and containment are reflexive on classes: AC A,

(2.17) (2.18)

ADA.

Ways to express an inclusion: (2.19) (2.20) (2.21)

ACB &BDA, ACB » B C A~, ACB

(2.22) (2.23) (2.24)

iCB ACB ACB

Inclusion and containment are antisymmetric: (2.25) (2.26)

i A

9. CALCULUS OF CLASSES

Transitivity laws: (2.27)

ACBABCC^-ACC,

(2.28)

AcBABCC

(2.29)

ACB AB
Union and intersection are monotonic: (2.30)

ACB ^AnC

CBnC,

(2.31)

ACB ^AUC

CBUC.

(2.32) Union and intersection are associative: (2.33)

AU(B\JC) = (AUB)\JC,

(2.34)

An(BC\C)

=

(AnB)nC.

Union and intersection are commutative: (2.35)

AUB = BUA,

(2.36)

AnB = BC\A.

Union and intersection are idempotent: (2.37)

Al)A = A,

(2.38)

AnA = A.

Union and intersection distribute over each other: (2.39)

An(B\JC) = (AnB)\J(AnC),

(2.40)

AU(BnC)

= {AUB)n{AUC).

Absorption laws: (2.41) (AnB)UB = B, (2.42) (AUB)nB = B. Duality laws and "De Morgan's laws": (2.43)

Al)B = AHB,

(2.44)

AnB = AUB,

(2.45)

iUB = IflB,

(2.46)

AnB = AUB.

Complementation is idempotent: (2.47)

~A = A.

46

2. SET THEORY

Laws involving difference: (2.48) (2.49) (2.50) (2.51) (2.52) (2.53) (2.54) (2.55) (2.56) (2.57) (2.58)

A~B = AU B , AUB = AU(B~A), (A\JB)~C=(J. L~C)U(B {AnB)~C={/ L~C)n(B (AnB)~C = A A~(BUC) = (/. L~B)n(A A~(B\JC) = (/. L~B)~C, A~(BUC) = (/L~C)~B, A~(BnC) = (/. L~B)U(A A~(B~C) = (/. L~B)U(A nc), A~{B~A) = A.

Laws involving the universal and empty classes: (2.59)

A n (B~ A) = 0,

(2.60)

4 n l = 4 ~A = 0,

(2.61)

4 U A = V,

(2.62)

A U 0 = A,

(2.63) (2.65)

,4n0 = 0, A~0 = A, 0~A = 0,

(2.66)

A U V = V,

(2.67)

J\ 1 1 V — / i i

(2.68)

J4~

(2.69)

V~ A = A

(2.64)

V = 0,

Laws for the ai lalysis of cases: (2.70)

A={/. inB)u(A

(2.71)

A=(AUB)n(A

.UB).

Laws involving symmetric difference: (2.72) (2.73)

(2.74) (2.75) (2.76)

AQ B = (Al)B)~(AnC) J4e(BeC)

= (4eB)eC,

AQB = BQA, A e A = 0, 4n(BeC) = (AnB)e(4nC),

10. AXIOM OP UNORDERED PAIRS

(2.77)

A e 0 = A,

(2.78)

I

(2.79) (2.80) (2.81) (2.82)

AUB = AU(AQB)

(2.83)

IeB = AeB,

(2.84)

AeBC(AeC)U(CQB),

(2.85)

(AUfl)e(CUD)C(4eC)U(Befl).

In what follows there are very few explicit references to Th. 12, and those are mostly confined to steps for which there is little or nothing else to say. 10. Axiom of Unordered Pairs The Axiom of Unordered Pairs states that, given any (possibly identical) sets a and b, there exists a set c whose only elements are a and b. f2 861

VoVfc(a is a set A b is a set => 3c(c is a set A yx(x G c O x = a V x = &))).

This axiom has an obviously weaker but interesting consequence, that if a and 6 are sets, then they are elements of a common set. Another consequence is that any class whose only elements are a and b must be a set, because any such class must be equal, by the Axiom of Extensionality, to the set whose only elements are a and b. The Axiom of Extensionality legitimizes the use of the notation "{a, 6}" for the (necessarily unique) set c whose elements are a, 6, and no other sets, and "{a}" for the unique set whose sole element is a. In the presence of the Axiom of Extensionality, the Axiom of Unordered Pairs can be written as follows. For all sets a and &, {a, 6} exists and is a set. Theorem 13. For all classes C and all sets a and b, x e {a, b}

x = a V x = b,

C = {a,6} » yx(xeC

» x = a V x = 6),

A class C is an ordered pair (or, simply, pair) with left side A and right side B iff C is a set A ^D^E(VX(X

(2M^

e C & x = D\/x = E)A

\fm(x (ED « x = A) A V K ( a ; e £ <& x = AVx = B)).

48

2. SET THEORY

An ordered pair is required to be a set and it follows from (2.87) that the left and right sides of an ordered pair must be sets. In the presence of the Axiom of Extensionality, the definition of ordered pair can be restated as follows: c is an ordered pair with left side a and right side b - c is a set A c = {{a}, {a, b}}. By the Axiom of Extensionality justifies standard notation for ordered pairs. For any sets a and b, let (2.88)

(a, 6}:= {{a}, {a, 6}}.

Again, if (a, b) exists, then a and b are sets. Each of them is an element of an element of (a, &}. Therefore, if either A or B is a proper class, then (A,B) can not exist. Introduce notation for ordered triples, quadruples, quintuples, and sextuples as follows. For all a, 6, c,d,e,f € V let (2.89)

{a,&,c} :={{«,6),c),

(2.90)

{a,b,c,d) : = {{a,b,c),d) =

(2.91)

{a,b,c,d,e) : ={{a,b,c,d),e) =

(2.92)

(((a,b),c),d), {{((a,b),c),d),e),

(a, b, c, d, e, /} := «a, b, c, d, e), /} = {{{{{a, 6} , c) , d), e) , / } .

The exact form of the definition of ordered pair does not matter for many purposes. The one chosen here is due to Kuratowski [125]. Our development of set theory is based on its special form. Regarding its complicating effect on the axiomatization, see the remarks in Tarski-Givant [240, p. 186]. The next theorem states the key properties required of any proper concept of ordered pair as well as ones specific to Kuratowski's definition. Theorem 14. For all sets a, b,c,de V, (2.93)

(a, &} e V,

(2.94)

(a,b) = {{a}} O a = &,

(2.95)

(a, &} =
The Axiom of Extensionality has been used to shorten the previous theorem. A more verbose but precise statement that ordered pairs exist is that is a set A B is a set => 3o(C is a set A

Vx(% E E <& x = AV z = B)))). It follows from the Axiom of Unordered Pairs that ordered pairs of sets exist. However, (2.96) is actually equivalent to the Axiom of Unordered Pairs. Moreover, the equivalence of these two statements does not depend on any axioms at all.

11, AXIOM OP RELATIVE PRODUCT

49

11. Axiom of Relative Product In the next two definitions we make use of the Axiom of Extensionality and a new notational convention. The notation {{x,y} : tp(x,y)} should be understood as a name for the (possibly non-existent) class of ordered pairs of sets x and y such that tp(x,y). If {{x,y} : tp(x,y)} exists, then (2.97)

c e {{x, y) :
With the help of this notational convention we say that a class C is a relative product of classes A and B if C = {(x,z) : 3v((x,y) eAh(y,z)e B)}, and that C is a relative sum of A and B if <7 = {{s;,.s):V,,(yisaset => ({x,y) £ A V (y,z) e B))}. Whenever the relative product and relative sum of A and B exist, they are unique by the Axiom of Extensionality. Denote the relative product and sum of A and B by A\B and A'fB, respectively.

(2.98) (2.99)

A\B := {{x,z) : 39(fay) eAA(V,z)e B)}, A t B : = { { M > : V , ( » £ V = > «sn,y> G A V (y,z) G B))}.

The Axiom of Relative Product guarantees the existence of the relative product of any two classes A and B. = A\B), or, simply,

An immediate consequence of the Axioms of Complementation and Relative Product is the existence of the relative sum of any two classes. Indeed, we have VAVS(^f-B exists). since Af B = V 2 ~ 3 | B . The notation "|" comes from Whitehead-Russell [255], while "f" originates with Peirce [194]. The operation o of composition is defined by (2.100)

AoB:=B\A.

We presented the Axiom of Relative Product only in its compact form, making use of the Axiom of Extensionality together with the special notation for ordered pairs and classes of ordered pairs simplify its statement. In the absence of Axiom of Extensionality the Axiom of Relative Product should be written in the following

BO

2. SET THEORY

form. VAVB3OVS(S € C

» 3v(s £ V)A

3p3,3r(Ve(e £ p « e = i ) A Ve (e £ g « e = «Ve = j/) A Ve(e € r « e = p V e = g) A r g i ) A 3p3,3r(Ve(e € p « e = z) ^'

^

A V c ( e 6 ? « e = j V e = !/) A Ve(e € r « e = pV e = q) Ar e B) A 3p3g3r(Ve(e £ p # e = i ) AV c (e£g

> e = i V e = 2)

AVe(e £ r

> e = j ) V e = ? ) A r £ C)).

Recall that Tarski's S3 expresses the existence of A-1\B. This long form differs from Tarski's S3 only in the placement of a few variables. Are (2.101) or S3 equivalent to significantly shorter sentences, or can they be replaced, in the context of other axioms, with substantially simpler axioms? We may now define the unit relation V2 as simply the relative product of the universal class with itself. V2 := V|V.

(2.102) Then V2 =

{(x,V):x,yeV}.

A class R is a binary relation iff it is a class of ordered pairs iff R C V2, and R is a binary relation on the class A iff R is a binary relation and the left and right sides of every ordered pair in R are elements of A, that is, Vc(c £ R => 3^(1 £ A A 3j,(p 6 A A c = (m,y)))). For every class R, R(~\V2 is called the relational part of R. It is the largest binary relation included in R. Note that A\B and A f B are always binary relations, although they are not necessarily sets. Theorem 15. Let R and S be arbitrary classes. (i) The relative product R\S and the relative sum R^ S exist and are binary relations: (2.103)

R\S C V2,

(2.104)

RjSCV2.

(ii) R is a binary relation » R C V2 « R C R\V « R C V|i2, (iii) V2j iZnV 2 , and 9 are binary relations. Various elementary results concerning relative sums and products are gathered in the following theorem. The notation is conventional—unary operations take precedence over binary ones. Among the binary operations, the order of precedence is |, f, n, U, and then ~ . For repeated binary operations, use association to the left, so that, for example, AUBUC = (AUB)UA, Several parts of the

11. AXIOM OF RELATIVE PRODUCT

Bl

following theorem have been given descriptive names or labels for later reference. For example, "|-mon" stands for "monotonicity of |". Theorem 16. Let R, S, T, X, Y, Z be arbitrary classes. The universal class may be used in place of the unit relation. (2.105)

R\S =(Rf] V2)\(S n V2) = (RC\ V2)\S = R\(S n V2),

(2.106)

R\\/ = R\\/2,

(2.107)

\/\R = \/2\R,

(2.108)

R^S = (i?nv2)t(5*nv2) = (RnV2)fs = Ri(snv2),

(2.109) (2.110) Relative multiplication is normal, and relative addition is dual-normal. (2.111)

0|fl = 0 = .R|0,

(2.112)

M]R = V2 =R\M.

Laws of properties.

(2.113)

s | r n v | i ? = s|(v|i?nr),

(2.114)

R\\/nS\T=(SnR\\/)\T,

(2.115)

i?|VnV|S = i?|V|5',

Schroder's Law [215, p. 150,152].

Tarski's Law [226] R\V = V2 Peirce's Remarkable Property [194]. 01-Rt 0 "

either

0 or V2,

(0f-R)|V is either 0 orV 2 , 0f-R|V is eitfjer 0 orV 2 , V|i?|V is eitter0 or V2. (|-assoc, f-assoc) Relative multiplication and relative addition are associative. (2.116)

fllOSiT)

= (R\S)\T,

(|-dist) Relative multiplication distributes over union. R\(SUT)

= R\SUR\T,

(SUT)\R

=

S\RUT\R.

(f-dist) Relative addition distributes over intersection. (2.118)

2. SET THEORY

(2.119)

(S

(duality) Relative multiplication and relative addition are dual. 2

(2.120) (2.121)

(|-mon, f-tnon) Relative multiplication and relative addition are monotonic. (2.122)

R C 5 =4- R\T C S\T,

(2.123)

R C S =j- T|i? C T|S,

(2.124)

ECS =s- EfTCSfT,

(2.125)

E C S =s-

TjRCTjS.

(Peirce's Law) "Two formulm so constantly used that hardly anything can be done without them" Peirce [194]. (2.126)

R\(S^T)CR\S^T,

(2.127)

(RjS)\T

C Rf S\T.

Miscellaneous interesting laws. (2.128)

R\Sn(R}T)=R\(SnT)n(RjT),

(2.129)

R\Sn(R1[T)=R\(S~T)n(R^T),

(2.130) (2.131)

R\Sn(X^Y) C(RnX)\SUR\(SnY), R\s\Tn(x^YfZ) c

One of the "miscellaneous interesting laws", namely (2.130), is used in the proof that a relational ideal J on a proper relation algebra gives rise to a congruence relation (see Th. 111). 12. Axiom of Converse The Axiom of Converse says that for every class A there is a binary relation whose elements are all the ordered pairs obtainable from ordered pairs in A by interchanging left and right sides. VA1BVX(Z

G B O- x is a set A ^V3z{x = (y,z) A {z,y) £ A)).

The uniqueness of the converse of a class follows from the Axiom of Extensionality, so define the converse of A to be the class (2.132)

A-1:={{Vlz):{3;,V)EA}.

Note that A-1 C V2 whenever A-1 exists. With this notation the Axiom of Converse can be shortened to

or simply " 1 exists).

12. AXIOM OP CONVERSE

53

Elementary results involving converse with relative sums and products are collected in the following Th. 17. Several parts of this theorem have been given descriptive names for later reference. Almost all parts of Th. 16 above and Th. 17 below have abstract algebraic counterparts that hold in all relation algebras. Converting between these two settings requires some care, because we are dealing here with classes of sets, rather than classes of ordered pairs. For example, the converse of (2.143) can fail, for if R and 5 contain no ordered pairs, then i?" 1 = S " 1 = 0, but this does not imply R C S, which would indeed be the case if R and 5 consisted exclusively of ordered pairs. T h e o r e m 17. The following statements hold for any classes The converse of a class is a binary relation.

R,S,T,P,Q.

i ? " 1 C V2.

(2.133)

(~1~1) Conversion is an involution on binary relations. (2.134)

OR" 1 )" 1 = -Rn v2>

R'1 = (fln V 2 ) ~ \

Conversion preserves the universal relation and the empty relation. (2.135)

V" 1 = ( V 2 ) " 1 = V 2 ,

(2.136)

0" 1 = 0.

(I" 1 ; t " 1 ) Laws connecting conversion with relative multiplication and relative addition (2.137) (2.138)

CRIST1 (RjSy1

=S~1\R-\ =S~1iR~1.

Conversion distributes over union, intersection, difference, and symmetric difference. (2.139)

{RUSy1 = R~1US~1,

(2.140)

(Rnsy1

= R~1ns~1,

(2.141)

{R^Sy1

= R~1~S~\

(2.142)

(Resy1

=

(

R~1QS~1.

)

Conversion is monotonic.

(2.143)

RCS => R'1

CS~\

Conversion commutes with relational complementation. (2.144)

R'1 = (V 2 ~ J R)" 1 = V 2

2. SET THEORY

Laws of properties (compare (2.113)—(2.115)^) (2 145) (2 146)

(RnV\S)\T = ( Z? 1 1 \ / l G\ 4- T 7 (XL LJ V J 1 1 J-

B|(5 |vnr), flKS-MVUT).

( T h . K) De Morgan's Theorem K [65]. (2 147)

R\SCT

^

(Bno) Forms of Tarski s axiom Rio-

(2 149)

SD^Iiilii"1, RIR-^SC 5D5|i?-!|B,

(2 150)

R-^C

(2 148)

R~X\R\SC

SDWR\R-\

(2 152)

RIR^ISC D R-^iRjS)^ i5C

(2 153)

B|(iFit5)C 5 3 (5tiFi)|ii,

(2 154)

R^\WS)Q 5D(5fii)|iFi, R\(R-^S)c 5D(5fB-1)|B,

(2 151)

(2 155)

-J D

|xt,

—^ ^ C 4|- JXjiJX D\lD— , — -1 l^O

La ws related to Peirce',> Law and transitivity. R^TD

(2 156)

(fltS-^lCSfT),

(2 157)

R\TC

(2 158)

(iit^KSf-^r), R\T C RIS-^SIT, (Ri~S^)\(SiT), ii|TC R\S=?iS\T, (RtS)\(S=?tT), ii|TC RlStS-^T.

(2 159) (2 160) (2 161) (2 162) (2 163)

Formulas for equation-solving. If R,S,r.r e v (2 164) (2 165)

2

, then

> SCR-ijT

« iJCTfS- 1

RCSjT ^ > 5-i|iiCT «

RT-iCS.

Tarski's equivalences. (2 166)

1

husnT-

= 0 <> ^ TIROS'1 =0 « 5|TnB"

12, AXIOM OP CONVERSE

Cycle law.

=T|S~1niE (2.167)

(rot) Modular laws and rotation. (2.168)

R\SnT=(Rf]

(2.169)

R\Sf\T =

(2.170)

R\SnT =

(2.171)

R~1\SnT =

(2.172)

i?|S"1nT =

n irx|T) n T, n T, n T, n R\T) n T, 1 nr.

Zigzag laws. (2.173) (2.174) (2.175)

1

P

T.

(2.176) Of course, the laws in Th. 16 and Th. 17 can be proved by appealing to the definitions of the operations involved, but many of them can also be given equational proofs that rely on earlier laws. For one of the miscellaneous laws from Th. 16, first note that, by various distributivity laws, we have R\S

n (x f Y) = (R n (x u x))\s n (x f Y) c((Rr\X)\sn(x^Y))u(iznx)|s.

However, rot

c (i?nx)|(5nx x;y1 c(ijnx)|(snf)

~ -mon, |-mon Rio

l-mon so, combining this with the previous equation, we get

c (Rnx)\suR\(Sr\Y).

56

2. SET THEORY

So far we have listed many laws whose abstract relation-algebraic versions are provable from the relation algebra axioms R1-R5 and R7-R10, the ones that do not involve the identity element 1'. These laws could therefore be said to belong to the "calculus of relations without identity". To construct a set-theoretical identity relation we need to upcoming Axiom of the e-Relation. 13. Axiom of the e-Relation The Axiom of the e-Relation asserts the existence of a binary relation consisting of all ordered pairs for which the left side is an element of the right side. 3EVx(x

e E O 3a3b(x

= {a,b)Aae

b)).

A less formal statement of the Axiom of the e-Relation is that {(0,6) : a £ 6} exists. A more formal statement of the Axiom of the e-Relation that does not rely on the Axioms of Extensionality and Unordered Pairs is 3EVc(c£E O 3a3b(a£b A 3y3z(yx(x€c

O I = I / V I= Z ) A

Vz (x e y <=> x = a) A

Vx{x E z O i = a V i = 6)))). The Axiom of the e-Relation implies the existence of a class that is unique by the Axiom of Extensionality. Let E be this class, that is, (2.177)

E:={{a,b) : o £ fo},

and let (2.178)

Id : = ( F r T t E ) n ( E - 1 t E ) ,

(2.179)

Di := E ^ E U E ^ E .

E is the e-relation, Id is the identity relation, and Di is the diversity relation. We expect E, Id, or Di to be proper classes. Reasons for their names and a few simple laws concerning the identity and diversity relations are given in the next theorem. For example, Id is the class of pairs of identical sets, while Di is the class of pairs of distinct sets. Note that Id is the intersection of the superset and subset relations on sets. Theorem 18. (2.180)

E=

(2.181)

Id = {(a, a):ae\/}

(2.184)

= {{{a}} : a € V},

2

(2.182) (2.183)

{(a,b):a£b},

Di = V ~ l d ={{a,b) E ~ T t E = {(a,6> 1

-aCb},

E " t E = {(a,6) : a D &}.

:a^b},

13, AXIOM OP THE e-RELATION

Id and Di partition the universal relation. (2.185)

Id U Di = V2,

(2.186)

Id n Di = 0.

Id and Di are symmetric. (2.187)

I d " 1 = Id,

(2.188)

D i " 1 = Di.

Id is an identity for relative multiplication, and Di is an identity for relative addition. (2.189) (2.190) Subclasses of Id are symmetric relations. (2.191)

R C Id =s- R'1 = R.

(Id-laws) Identity laws. (2.192)

Id cTF^jR,

(2.193)

Id C R-1 f S ,

(2.194)

Id C

(2.195)

Id C

(Di-laws) Diversity laws. (2.196)

R'\R~1 C Di,

^o 1 nT^

D

o— i z^- rv

{z.Lui)

ii\ii

(2.198)

R^IRC Di,

\— Ul,

(2.199) iJ^'llcDi. For any classes A and B, let A^ B = A\B~1. With the addition of the Axiom of the e-Relation (which entails the existence of Id) it becomes possible to replace the Axioms of Relative Product and Converse with a single axiom that guarantees the existence of A f B, namely the sentence 5s in Chapter 2, §3. The converse and relative product can then be obtained as follows. 1

A\B =

=\d\A, 1

1

88

2. SET THEORY

14. Axioms of the calculus of relations In the last few sections, the Axioms of Unordered Pairs, Relative Product, Converse, and the e-Relation have been added to the list of axioms that give rise to the calculus of classes, namely, Extensionality, Empty Set, Intersection, and Complementation. Here is a complete list of the axioms at this point, together with some of their consequences. Extensionality : membership determines identity, Empty set : 0 exists and is a set, Complementation : R exists, Intersection : R n S, R U S, and the universal class V exist, Pairs : the sets {x,y} and (x,y) exist for all sets x and y, Relative product : R\S and the relative sum R'f S exist, Converse : R~x exists, e-relation : the class E exists, so the classes Id and Di exist, The Peirce-Schroder calculus of relations of the nineteenth century is concerned primarily with the empty set, complementation, intersection, union, ordered pairs, relative product, relative sum, converse, the unit relation V2, the identity relation Id, and the diversity relation Di. The Axioms of Extensionality, Unordered Pairs, and the e-Relation are products of the twentieth-century set-theoretical setting. 15. Kinds of relations The class R is said to be symmetric iff R~x C R, i.e., VssVj^a;, y) 6 R~x => {y,x) G R)- A class may be symmetric without being a binary relation. For example, every class that contains no ordered pairs is symmetric. Theorem 19. (i) 0, Id, Di, and V2 are symmetric binary relations. (ii) For every class R, R'1 n R, R'1 U R, R~X\R, RIRT1, R'1 f R, and Rf R~x are symmetric. (iii) If R and S are symmetric classes, then Rtl S and RU S are symmetric classes. The binary relation R~* n R is the largest symmetric relation included in the class R, and it is called the symmetric part of R. If R C V2, then the relation R~x U ii the smallest symmetric relation containing ii, and is called the symmetric relation generated by R. A class R is transitive iff R\R C iZ, and R is a transitive relation if R is transitive and R is also a binary relation. A class may be transitive without being a binary relation; any class that contains no ordered pairs is transitive. We say that a class R is dense iff R C R\R. Since R\R C V2 for every class R, a dense class must be a binary relation. Theorem 20. (i) 0, Id, Di, and V2 are transitive dense relations. (ii) The following statements are equivalent. R is a transitive binary relation,

15. KINDS OP RELATIONS

R\R C R C V3,

RCR-ifR. PROOF. Proof of (ii): The equivalence of the first two conditions holds by definition. For the proof of the equivalence of the other parts, note that the last two conditions imply fiCV2, take i i = S = T in (2.164) and get R\RCRCV2 «- RClf^fR «- RCRfW1. D Peirce [199, 4.94] observed in 1893, Yet really, the form If? is all-important, inasmuch as it is the basis of all quantitative thought. For the relation expressed b j it is transitive. . . . This is not only a transitive relation, but it is one which contains the identity under it. . . . But it is further demonstrable that every transitive relation which includes identity under it is of the form If I. (Peirce [199, 4.94]) We formulate Peirce's theorem as follows. Theorem 21. Let R be any class. (i) RfR-1 is transitive and Id C RfR-1. (ii) The following statements are equivalent: R is a transitive binary relation and Id C R,

P R O O F . Part (i): By (2.194), RfR-1 contains Id. For transitivity, we have (Rf R-^KRf R-1) C RfiR-t^fR 1

C Rf Di fTF T

= RfRr

Peirce's Law, f-mon (2.198), f-mon (2.190), (2.108)

Part (ii): By part (i), if R = Rf R-1 then i i is a transitive binary relation that contains the identity. For the other direction, assume R\R C R C V2 and Id C R, We get R C RfTF1 by Th^20. For the opposite inclusion, from Id C R get V2 n R C Di, hence V2 n R-1 C Di by -1-mon,_(2J.44), and (2.188). Relatively add R on the left of this last inclusion, to get R f R-x = R f(V2 n R~1) C R f Di = R. D Although Peirce's theorem was stated only for RfR-1, it is also true for R- f R, i j " 1 fR, and RfR~1. Here is a corollary that embodies this observation. The list of equivalent statements can be considerably extended. In this connection see Schroder [215, p. 338] and Ng [187, p. 2]. 1

2. SET THEORY

Theorem 22. For any class R, the following statements are equivalent: R\R\J\d CRC V2,

A class R is said to be an equivalence relation if R C V 2 ,ii is transitive, and R is symmetric. Theorem 23.

(i) 0, Id, and\/2 are equivalence relations. (ii) // R and S are equivalence relations, then R(l S is an equivalence relation. The next theorem lists several different ways, drawn from Tarski [227], TarskiGivant [225], and Chin-Tarski [49], to assert in the calculus of relations that a class is an equivalence relation. See Th. 308, where these conditions are all shown to be equivalent in an arbitrary NA. Theorem 24. For every class R, the following statements are equivalent: (i) R is an equivalence relation, (ii) R\RC RCR-1, (iii) R\R = R = R-\ (iv) R\R-X = R, (v) R~1\R = R, (vi) R C V2, RIRT1 £ R, and R'^R C R, (vii) RCV2, R\R C R, and R\R C R. PROOF, (i) => (ii): Assume that R is an equivalence relation. Then, by definition, R is a binary relation which is transitive and symmetric. This means that R C V2, R\R C R, and R~l C R, so we need only note that

R

= R n v2 (2.134) (2.143)

= (R ) c R-1 (ii)=^(iii): Assume R\RC RCR-1. Then R-1 Q(R-

r1 V

=R

2

(2.143) (2.134) (2.133)

Combining this with the assumption that ii C i i " 1 gives i i " 1 = R. Also, X

\R

(2.174)

15. KINDS OF RELATIONS

61

C R\R\R

(2.122), (2.123)

C R\R

(2.122)

From this inclusion, together with the assumption that R\R C R, we conclude that R = R\R. Thus (iii) holds. Obviously (iii) implies (ii), (iv), and (v). Because of (2.133), (iii) also implies (i) and (vi). Therefore (i)-(iii) and equivalent, and each of them implies (iv), (v), and (vi). (iv) => (iii): Assume R\R~X = R. Then R C V2 by (2.103), so ii" 1 = (RlR'1)'1

hyp.

= (ii"1)"1^"1 = R\R~

(2.137)

X

(2.134)

= R

hyp. -1

From this and the hypothesis we get R\R = R\R = R. Thus (iii) holds. By a very similar calculation, (v) implies (iii), so we have shown that (i)-(v) are equivalent. (vi)=>(i): Assume R C V2, R\R-1 C R, and R~*\R C R. The latter two inclusions do not imply that R is a binary relation, which is why (vi) includes an explicit statement to that effect. First we obtain itT 1 C R as follows. R'1 C R-^iR'1)'1^-1

(2.174)

1

(2.134)

= R~ \R\R~

1

C R\R-* CR

(2.122)

From R~ * C R C V2 we derive R = R~ *, as we did in this proof that (i) implies (ii). From this last equation and either the second or the third hypothesis, we conclude that R is transitive. Therefore (i) holds. We now know that (i)-(vi) are equivalent. (iii) => (vii): Assume R\R = R = i?" 1 . Then R C V2 by (2.103), and (2.169) = R\(RnR)

hyp.

= R\®

(2.60)

= 0

(2.111)

so R\R C R~ by (2.21) and (2.47). By a similar calculation we also obtain R~\R C R~. (vii) => (vi): Assume R C V2, R\~R C S, and B|i? C S. Then ^

^

(2.171) (2.123) (2.60) (2.111)

62

2. SET THEORY

so i r 1 R C R, and D— f ( D r~i D1 D \ 1 D — ^ RIR-1 r-,1 JX ^— ^iT 1 1 iX\ILJ -T c (flnfl)lfl"1 = 0 -R"1 =0 X

so R\R~

C R.

(2.172) (2.123) (2.60) (2.1H) D

For a given equivalence relation R and an arbitrary set a; € V, we will define x/R to be the equivalence class of x, that is, the class of sets that are in the relation R to the set x. We do not yet have enough set-theoretical axioms to prove the existence of such a class for every R and x. Even if x/R does exist, it need not be a set. For example, V2 is an equivalence relation under which all sets are equivalent to each other, so the equivalence class of every set is the universal class V, which may not be a set. The existence of the equivalence class x/R follows from the Class Existence Theorem; see (2.390). In case the equivalence relation R is also a set, we can eventually prove that x/R not only exists but is a set. This conclusion is encapsulated in (2.420).

16. Coextensivity Next we define a special operation on couples of classes that yields a "coextensivity" relation. For all classes R and 5, let (2.200)

RoS := (RjS)n(RjlS).

In expressions involving o and |, perform | first, followed by o. When o shows up with fl, U, f, and other binary operations, we tend to use parentheses for clarity. An ordered pair of sets (a;, y) is in the relation RoS iff the .R-image (see p. 94) of the left side x coincides with the S^-image (or S-preimage) of the right side y. A restatement of this characterization and some other basic properties of o are given in the next theorem. The first part shows that o will be obtained from the " in (2.99). definition of f when "V" is replaced by Theorem 25. For all classes E, R, S, and T, (2.201) (2.202)

RoS = {<*, y):V*{{x,z) Id = Id .Id, c

(2.203)

Id = E " ] oE 1

(2.204)

RoSC V 2 ,

(2.205)

V2~(fl<>S) = R\SUR\S,

(2.206)

RoS = Ro

(2.207)

(Rosy

(2.208)

Id

(2.209)

Id CR~

I

_

s, S^oR'1,

CRoR- I

16. COEXTENSIVITY

(2.210)

'RoR~1CD\,

(2.211)

H^oflCDi,

(2.212)

RoR-1 C Di,

(2.213)

R'1 oRC Di,

(2.214)

R~1\(RoS)CS,

(2.215)

(.RoS)^" 1 Cfl,

(2.216)

i F ^ - R o S ) CS,

yZ.Zlt)

\Tt O J ) \ j

(2.218)

RoR~ and R~ oR are transitive and symmetric,

(2.219)

if R is transitive and symmetric, then R\(RoS) C RoS,

(2.220)

ifR is a transitive relation then {E~l o R^R^R'1

(2.221)

(RoE^iE'^SoT) C R\SoT,

(2.222)

(R

(2.223)

(RoE)\((E-1:\S)oT) C

(2.224)

(iZo(5t-E))|(£;- 1 oT) C

(2.225)

(iioBJKtr'tS)*?) C (RjS)oT,

(2.226) (

i

^_ XL,

o E) C '

1

i

(2.227)

(R

(2.228)

( i i o S ' ^ K S o T ) C ~R~oT = RoT.

PROOF. (2.201): By (2.200). (2.202): Id -o Id = (Td"f Id) n (Id f Id")

(2.200)

2

2

= ((v nid")fid)n(idt(v nid"))

(2.108)

= (Di f Id) n (Id f Di)

(2.5), (2.182)

2

2

= (id n v ) n ( i d n v ) = ldnV = Id

2

(2.190) (2.38) (2.185)

(2.203): By (2.178) and (2.200). (2.204): By (2.200) and (2.104). (2.205): (2.200) (2.56)

64

2. SET THEORY

= R\S U R\S

(2.47), (2.1.21)

(2..206): R

oS=(R}S)n(R}S) = CRfS)n(i!ts)

(2.200) (2.36)

= (If5)n(flff)

(2.47) (2.200)

(2..207): 'to

.200)

'to

((^t^nc^t^))-1

.140)

'to

(RoS)'1 ==

.138)

/o

.144)

1

^(Risy'niRiS)-

--(s-^R-^nis-^R- l) = (5" 1 t(v 2 ~ J R- 1 ))n((v :

1

r)

i. D—1\

= (5"1t-R-1)n(S-1t-R"') = (5- 1 t-R" I )n(S" I t-R" : = S~1oR-1

(2 .108)

'to

(2 .36) .200)

(2.,208)-(2.213): By (2.192)-(2.199). Proof of (2.214):

R~'1\(RoS)CR-1\(RjS)

(2.123) (2.152)

QS (2..219): First we have (2- 200) 'to

R\(RoS) = R\((R1 S) n WS)) C R\(Rf S) n R\(Rj~S) c (R\RiS)n(R\RiS)

J

12), (2.36), (2.123), (2.15)

(2. 126) , (2.30), (2.35)

We have R\R C R since R is transitive, so R\R} S C Rf S by (2.124). We also h_ave i?" 1 C B_since R is symmetric, so i ? " 1 ^ C R\R C R by (2.122), hence

~RnR-1\RCR~nR = ®. Then

^) = i?|0 = 0

(2.169) (2.111)

soR\RCRby (2.47), (2.5), (2.21). Hence fl|lf'S' Cflf-S*by (2.124). Combining these observations with the first part of the proof gives

R\(RoS) C (RlRjS) n (R\Rj~S) = RoS.

16. COEXTENSIVITY

(2.220): (E^oR^l IKiJ^oE) C (E-1 \ ROI-RK-R-1^) 1

)

C (E~ \R

|-mon Peirce's Law

c (S-^-R

R is transitive, t-mon, |-mon

C£-ifB|

Peirce's Law twice, t-mon, t-assoc

CS-ifDi

Di-laws

f£

= E~1\E Proof of (2.221): Calculate as follows: {RoE)\{E-1\SoT)

Peirce's Law twice, f —mon (2.149), t-mon = ~RjS}T

duality

and (RoE^E-^SoT) C (RfE~)\(E~1\S\T) 1

|-mon

C (R-\ E)\E~ \S'\T

Peirce's Law, |-assoc

= R\SfT

(2.152), |-,t-mon

so, combining these two results, we have / O y , IPW f IT* — 11 c 1 A T~!\ f— / D | C T~!\ r~\ ( T21 C 4- T ^ ^JXO£/J|^ii'

p O i ) ^ ^ r t | o j i J I I yri\O \ 1 ) —

p i n . ryi O 1.

For part (2.223), a similar calculation suffices. (R«E)\((E-^S)oT) C (RifEJKE-1 iSfT) — tt\ hj\yhj~

\ Dj \ 1

CfltStT

-mon D

eirce's Law twice, t-mon

itio, t-mon

{RoE)\{{E^\S)oT) (1 ( f? A- JPW ( r r 11 c + T 7 ^ - \-**- -C/ J I -C/ J iI — / p 4. p \ 1 p — 1 1 C1 4- T"1

duality

CB|StT

PLIO

= R~-\SfT

duality

Peirce's Law, -assoc

2. SET THEORY

Combine the previous inclusions to get (RoE)\((E-1^S)oT) C (R^S^l r)n(i?f5tr) = (R}S) For part (2.225),

(

C (R1[E~)\(E-1-\lS1iT)

|-mon

C~ T> 4- ~I?I ( 7 ? ~ 1 4- ~O\ 4- T^

Peirce's Law twice

C(flt5)tT

Rio, t - m o n

XL

O £ / I I I -C/

t j

1O 1 )

— f ^ Z ? 4 - ZT'^I/'ZT'— 1 1 C 4- Tl\ \ 1 JX S-J) \ S-J *^ \ -*- )

duality Peirce's Law, |-assoc

CR\SjT

Rio

CR^S^T

duality

Combine the previous inclusions to get {RoE)\((E-lj{S)oT) C (RjSjT)

oT.

Proof of (2.227): Use any one of (2.221)-(2.226), with 5 = Id or 5 = Di.

17. Functional and injective relations Suppose R is any class. A class A is said to be a domain of R iff every element of A is the left side of some pair in R and the left side of every pair in R is an element of A. A class B is said to be a range of R iff every element of B is the right side of a pair in R and the right side of every pair in R is an element of B. By the Axiom of Extensionality, a class can have at most one domain and at most one range. The class R is functional iff no distinct ordered pairs in R with the same left sides have distinct right sides, or, more formally, R is functional <S> V^V^a;,?/) <E R A (x,z) € R => y = z). (2.229) We say that R is a function iff R is a binary relation and R is functional. We write R : A —> B just in case R is a function, A is the domain of R, and the range of R is a subclass of B. These definitions are revisited on p. 121. T h e o r e m 26. R is functional

iffRClR\D\

= 0 iff R-1\R

C Id.

PROOF. Starting from (2.229), we have R is functional <=>VxVyVz({x, y) 6 R A (x, z) 6 R => y = z) ^ f o . a ; ) G R-1 A <x,«) G R => {y,z) € Id) z(3x({y,x) ,«»,*> G I-RC Id

e R'1 A {x,z) e R) => {y,z) e Id) f => G Id)

17. FUNCTIONAL AND INJECTIVE RELATIONS

Id then

RDR\D\ C R\(D\ DR'^R)

rot

C i2|(Di D Id) = JR|0

|-mon

= 0.

If Rr\R\D\ = 0 t h e n i i C R\D\, so i C DfnV

|-mon (2.103), (2.148)

2

= Id.

D We say that a class R is injective or one-to-one if RT1 is functional, and that R is a bijective if R is functional and injective. If R is a bijective binary relation, we call it a bijection. If R is an injective binary relation, we call it an injection. A class that is functional, injective, or bijective need not be a binary relation. For example, if then R is bijective, but R is not a binary relation because two of its elements are not ordered pairs. Injectivity can be characterized in a manner similar to the way functionality is characterized in Th. 26. Consequently bijectivity may be expressed in several ways. T h e o r e m 27. R is injective

= 0 iff R^R'1

iffRr\R\D\

C Id.

See (2.394) and Th. 62 for an application of the following theorem. Theorem 28. For all classes R and S, if S C Ro Id o r S C i J o E then S is ', and if S C Idoi? or S C E o R then S is injective. PROOF.

By (2.202), (2.203), (2.207), and (2.227), if S is either Id or E, then S) = (S'1 o R-^KRo

(Ro Sy^iRo

S)

CS~1oS=\d,

so R o Id and R o E are functional relations. Subclasses of functional classes are functional. The rest of the proof is similar. Next we have two expected theorems, followed by a less obvious one. Theorem 29. If R is a functional class, then .Rl-R"1 is an equivalence relation. If R is an injective class, then R~1\R is an equivalence relation. PROOF. Assume R is a functional class, so that R-1\R C Id. Then R\R-1 is transitive because

^

1

^

1

1

and symmetric because C V2,

=R\R~\

68

2. SET THEORY

hence R\R-1 is an equivalence relation; see Th. 24. Theorem 30. If R and S are functional classes, then R\S is a function. If R and S are injective classes, then R\S is an injection. PROOF. If R and 5 are functional, that is, R-1\R C Id and S~1\S C Id, then R\S is a function since R\S C V2 and (2.137), |-assoc R is functional 1

= 5" |5 C Id

S is functional

Theorem 31. If R and S are functional classes, then i?f S is a function. If R and S are injective classes, then R^S is an injection. The associative law for relative multiplication is used in the proof of Th. 30. This usage is essential. The abstract equational version of Th. 30 (see Th. 398) holds in all relation algebras but fails in some semiassociative relation algebra. In contrast, the abstract equational version of Th.31 (see Th. 331) not only holds in all semiassociative relation algebras, but also holds in all weakly associative relation algebras (Maddux [143, Th. 14]). Therefore Th. 31 may be proved without referring to more than three sides of ordered pairs in R and S at any one time, but the same cannot be said for Th. 30: it is necessary to simultaneously consider four objects at some stage of the proof. In the next two theorems we see that if two functions have disjoint domains then their union is a function, while if two functions have the same domain and one is contained in the other, then they are equal. Similarly, the union of two injections with disjoint ranges is an injection; if one injection contains another and has the same range, they are equal. Theorem 32. If R and S are functional and R~1\S = 0, then R U S is functional. If R and S are injective iZ|5 - 1 = 0, then R\J S is injective. A s s u m e R-X\R

PROOF.

C Id, S ' ^ S C Id, a n d R'^S

S-X\R = S'^iR-1)'1

(2.230)

= 0. T h e n

= (R-^Sy1 = 0"1 = 0,

so R U S is functional because = (R-1US-1)\(R{JS) 1

1

1

"'-dist 1

= R~ \RUR~ \SUS~ \R[JS~ \S

|-dist

C Id U 0 U 0 U Id

R, S are functional, (2.230)

18. FUNCTIONAL AND INJECTIVE PARTS

69

Theorem 33. If R and S are functions, RC S, and R\\/ = 5*|V, then R = S. PROOF. Note that S C S|V by Th. 15(ii) R

c s = s n 5|v = S n R\V

hyp.

= Sni?|(ld U Di)

two cases

C(Sni?)U(Sni?|Di)

|-dist

ci?u(S'ns'|Di)

RCS

= RU0

S is functional

= R D

18. Functional and injective parts For every class R let (2.231)

~R:= Rn(R~i\d), —> and refer to R as the functional part of R. For every class R let (2.232)

R:=Rn(\~

and refer to R as the injective part of R. Much of the next theorem may be summarized by saying that the functional part of a class is a function, the injective part is an injection, a class is a function iff it is its own functional part, and an injection iff it is its own injective part. Theorem 34. Let R be an arbitraryclass. (2.233)

RURCRHV*

(2.234)

R = Ro\d,

(2.235)

*R=\doR, <-—i

(2.236)

=(R~1)^i

R

- ^ - 1 < - - 1

The functional part is a function: (2.237)

—> —>

(2.238)

iJ

(2.239)

R is a function,

(2.240)

R= R & R is a function.

|.RCId,

2. SET THEORY

The injective

part is an

injection:

(2.241) <-

<--1

(2.242)

i?|i?

Cld,

(2.243)

i? is an injection,

(2.244)

R = R t$ R is an injection.

—> P R O O F . For (2.233), first n o t e t h a t t h e inclusion RU calculus of classes. T h e relative s u m of any two classes is t h e intersection of a class w i t h a binary relation is again a

«— R
functional a n d injective p a r t s of any class are b i n a r y relations, i.e., R U R C V 2 . T h e following calculations are proofs of (2.234), (2.235), (2.237), a n d (2.238), respectively.

) = Ro\d, = Rn(\dJ\R) = ( D i t f l ) n ( l d t ! ) = (Td" f R) n (Id fR) = Id o R, RnR\D\ R

C R\D\ HR\D\ = 0,

IRCR-^iR^d)

C Id.

Proof of (2.239): R is functional by Th. 26 and either (2.237) or (2.238). R is therefore a function since the functional part of every class is a binary relation by (2.233). Proof of (2.240): If R = R then R is a function by (2.239). For the converse, assume R is a function, i.e., RC\R\D\ = 0 and RCV2. Then

R = RH\/2 = (JRnv2nJR|Di)u(i?nv2~i?|Di)

= R. The operation of forming the functional part is not monotonic, for if a, b, c 6 V are distinct sets, i? = {(a, 6}}, and S = {(a,b) , (o,c)}, then R C S but R = R and 5 = 0, so R g 5. Similarly, the operation of forming the injective part is not monotonic. The relation E is easy to express verbally. A set is in the relation E to another set if it is the sole element of the other set, i.e., E maps each set x to the singleton

set {x}.

18. FUNCTIONAL AND INJECTIVE PARTS

71

-s—

Theorem 35. E is a bijection, (2.245)

V2 = E|V = (ldoE)|V,

(2.246)

E={{x,{x}):x£\/}.

PROOF. E is a binary relation by (2.233), so E is a bijection by (2.235) and Th. 28. It follows from the Axiom of Unordered Pairs that E is denned on all sets, hence V2 = E|V, and V2 = Id o E by (2.235), so (2.245) holds. All the elements of E are ordered pairs of sets, and for any sets x and y, we have {x,y)€E

{x,y)e

En(ldfE)

<£> ( x , y ) e E / \ { x , y ) e

Id f E

«=> x e y AVa(a e\/ => x = a V a ^ y) « x e j / A V o ( a e V =^ (a € y => x = a)) <$ x e y AVa(a e y => a = x) ^

{x} = y

*> {x,y) =

{x,{x}).

a The next two theorems contain some other laws governing functional parts and injective parts. Theorem 36. If R is a class, then (2.247) (2.248) (2.249)

R-^R

C\d,

->-i

(2.250)

R

\RC\d,

—)

(2.251)

R = R,

(2.252)

R\R = (Rr)(Rn\/\R

)~")\R,

(2.253)

R\R = (Rr)(Rr)\/\R

)~")\R.

PROOF. The following calculations are proofs of (2.247), (2.248), (2.249), and (2.250), respectively.

RHR\D\ C (ijfld) ni?|Di C R\D\ n R\D\ = 0, RnR\D\ Cfln-RIDilDi" 1 C i ? n B = 0, ^CR)

C Id,

2. SET THEORY

R

\RC (flfld)

1

| B = ( l d t - R - 1 ) | - R C Id.

Proof of (2.251): By (2.239) and (2.240). Proof of (2.252): Let S := RC\ V\R (2.254)

. Then

#: ^

~

i?=(Bn5|Di)U(Bn5|Di) —>

two cases

—>

c (2.247)

Hence rot RCR (2.254)

c

(2.231) CR\R

l-mon

For (2.253), apply (2.252) to R, and invoke R = R.

D

T h e o r e m 37. If R is a class, then (2.255) (2.256) (2.257)

R\R

1

(2.258) (2.259)

CId, C Id,

R=

(2.260) (2.261)

R\R= (RH (RH V\R

)^)\R.

In general, relative multiplication does not distribute over intersection, but certain distributive laws do hold when some of the classes involved are functional or injective. T h e o r e m 38. Let R, S, and T be arbitrary classes. Then (2.262)

'

18. FUNCTIONAL AND INJECTIVE PARTS

(2.263)

S\RDT\R=

73

(SnT)\R =

If R is functional then (2.264)

R\(SnT) = R\SDR\T,

(2.265) If R is injective then (SnT)\R = S\RDT\R. PROOF. For part (2.262), calculate as follows. R\SnR\T C R\(SnR

\(R\T))

rot

C R\(Sn\d\T)

-assoc, (2.250)

= R\(Sr\T)

c

R\Sr\R\T

-mon

c

R\SDR\T

-mon

T h e o r e m 39. —¥—¥

(2.266)

r~ /'pi

—y pW

pi D r~ ( pi

D\~^

DID

—y —y

(2.267)

K\K ^ [K\K)

(2.268)

—f

pi p

( pi

4—4—

4—

,

p\~^

K\K — {it\K)

,

(2.269)

p i p (— ( p i p \ ^ ~ iX iX ^— ,

(2.270)

4— 4— p i p (— ( p i D\4— i T i T ^— i

(2.271)

Pi p / Pi p\^~ iX iX —

—

4—

PROOF. Proof of (2.266): Note that R\R C R\R by |-mon since R C R. Also note that from R\RnR\R\D\ = R\R n R\(R\D\)

|-assoc

= R\{RnR\D\)

(2.262)

= R\
(2.237)

= 0

|-normal

it follows that R\RC V2~iZ|fl|Di

=R\Rf\d.

2. SET THEORY

Combining these two observations yields R\R

C R\R n (R\Rj Id) =

By very slight modifications we also obtain proofs of (2.267) and part of (2.268), namely, R\R C (R\R)^.

For the other half of (2.268), we have

= R\RC\R\R\D\ CR\(RC\R

|i?|ii|Di)

rot.

CR\(RnR\D\)

(2.148)

—y —y

= R\R D It is easy to construct examples of relations R which show that the inclusions in (2.266), (2.267), (2.269), and (2.270) may be strict. Consider four distinct sets o, b, c, d € V and let R = {(a, b), (a, c),
but {(o,d» = R\R = R\R = (R\R)^ = (R\R)~>. The following theorem was known to Tarski; see Tarski [227, p. 108] and Tarski-Givant [240, p. 129f]. The proof given here is entirely computational and can be carried out in the abstract algebraic setting of an arbitrary relation algebra, which is essentially what is done in the proof of Tarski-Givant [240, 4.6(iii)]. Theorem 40. Assume R and S are classes. (i) / / R is functional then S fl ((5 U R) f Id) is a function. (ii) If R is injective then S fl (Id f(5 U R)) is an injection. P R O O F . First we p a r t i t i o n 5 fl ( ( 5 U R) f Id) into two p a r t s .

5n((5Ui?)fld) = ((5 n R) U (5 n ~R)) n ((5 U R) f Id)

two cases

= (5 n R n ((5 u R) f id)) u (5 n R n ((5 n R) f id))

n-dist

= s n R n {(s u R) t id) u (S u R)^ The first piece of the partition, namely Sr\Rn((SUR) f Id), is functional because it is included in the functional class R. The second piece of the partition, namely (5ni?)~ > , is a functional part and hence is functional. Both pieces are binary relations, and hence both are functions. Finally, these two functions satisfy the condition required in Th. 32 for their union to be a function. Indeed, we have

1(5 C\ R)

|-mon, '-mon

C D i n ( l d f ( S n f l ) ^KSTli?)

Di-law, f" 1 , (2.187)

= Di n Id

(2.153)

19. PROJECTION FUNCTIONS

The next theorem shows that if F be a functional class, then F U (Id ~(.F|V)) is a functional class that is defined everywhere it could be, in the sense that every set is the left side of an ordered pair i n F U (Id ~ F\\l). Theorem 41. If R is a functional class, then R\J (Id ~i?|V) is functional. P R O O F . Let S = R U (Id ~ R\V). Then

S~ \S C R

|(iZU-R|V) 1

~ -mon, |-mon

1

= R~ \RU(R~ \Rjj)

|-dist

C Id U 0

R is functional, Rio

Theorem 42. If R is functional, then R\V n (R\SoT)

C

R\(SoT).

PROOF.

R\Vn(R\SoT) rot |-mon C R\((R~L\R\S^T)

n (R~L\R\S^T))

Peirce's Law, |-assoc

C ii|((ld|5tT) l~l (5fT))

R is functional, |,f-assoc, (2.148)

D 19. Projection functions The main goal of this section is to prove the existence of these two binary relations: l \ \

x

i D l

1^1

x

i U

t

V

J ;

{{{x,y),y):x,yeV}. It will be shown in (2.306), (2.307), and Th. 47 that these binary relations can be obtained from E and Id by using only U, PI, ~ , |, and f. Our starting point is a construction due to Tarski.

2. SET

THEORY

Theorem 43 (Tarski [227, 3.9, p. 108]). Suppose C is any class. Let (2.272)

A :=

(2.273)

B:

Then A and B are injections and the following statements are equivalent: VxVy(xeVAyeV 1

=> 3z(Vw({w,z)

EC O w = xVw = y))),

2

A\B- =V . Tarski's construction, given in (2.272) and (2.273), is mentioned and applied several times in his 1942-43 manuscript Tarski [227, pp. 108, 111, 156, 209, 207, 220-221]. Its converse dual, involving the functional part instead of the injective part, is used by Tarski-Givant [240, 4.6(ii)]. One of Tarski's variations on this construction comes close to our goal, which is Th. 46 below. A class C is called extensional iff (C" 1 o C) fl C " 1 ^ ^ C Id. For example, E is extensional.

Theorem 44 (Tarski [227, p. 207]). Suppose C is a class. Assume (2.272), (2.273), and let (2.274)

Ai : = ^ n V | B n ( V | C t C ) ,

(2.275)

Si :=-BnV|An(V|CtC).

Suppose C is extensional. Then the following statements are equivalent: (x,z)£A1A(y,z)eB1,

(2.276) (2.277)

3u3v(Vw({z,w)

eC^w

= uVw = v)A

yw({u,w)

G C <=> w = x) A

yw((v,w)

£C<$w

= xVw = y)).

Since E is extensional, it follows from Th. 44 that if C = E then (2.276) and (2.277) are also equivalent to x = (y,z). Th.44 requires the assumption that C is extensional. That more complicated constructions would succeed even without the assumption of extensionality was stated by Tarski-Givant [240, p. 130]. Such a construction is given below. We begin by denning four special operations, denoted by superscript suits, that may be performed on classes to obtain new classes. For an arbitrary class C, let C*, C^, C*, and C* be the classes denned as follows.

(2.278)

C*:=(CnV|C

)">,

(2.279)

C^:=C*\C,

(2.280)

C :=(CnC*|Di)^,

(2.281)

C* := C\C U

C n (C\C ~ C ^ V ) .

These relations were designed to work properly under the assumption that C = E^ 1 . Here are illustrations of the meanings of the first three relations obtained in

19. PROJECTION FUNCTIONS

the only singleton

{

{x}

, . . . non-singletons (if any)... }

the only singleton

{

{x}

,... non-singletons (if any)... }

the only singleton

{

t h e Qnly

W

non

_singleton

^

}

^

^ y

Theorem 45. For every class C, (2.282)

C, C , C , and C are functional binary relations.

(2.283) (2.284)

C\C = (C*nC)\C = C^ n C*, C* is functional,

(2.285)

C*|C = C*|(CnC|Di),

(2.286)

C ^ K C n(C\C ^ C ^ I D i l C T 1 ) C C" 1 .

PROOF. Proof of (2.282): C, C*, and C* are functional binary relations because they are functional parts of classes. C^ is a functional binary relation because it is the relative product of two functional binary relations. Proof of (2.283): We have C\C = (C* H C)\C by (2.252), and (C* D C)\C C C*\C = C^ by |-mon, so C\C C C^, hence C^ n C\C = C\C. Also,

c* nc**^c\{c\c"C^y

c

^

^

Then 9

*

= c ' n (c\c u cn = (c^ n c|C) u (c^ n c^ic n (c|c = c\c.

—> —> Proof of (2.284): C\C is functional because it is the relative product of functional binary relations; see Th. 30. C^\C n ( C | C ~ C^)"* is functional because it is included in the functional class (C\C^C^)^. C is the union of these two functional binary relations, so, by Th. 32 it suffices to show that

First, we have

C(C*"'nC ^(Cn^lDi))

(2.280)

2. SET THEORY

CC*

1

|(C*|Di)n(C7 \C)

-mon -assoc, C is functional (2.250)

C ld|Di Old

Hence

(c\c) x\(c

r 1(C|C

) |-mon

(C<> C)

= ((c* n c7)|C)

(2.283), |-assoc

|C

^ —1

->-l

fl" 1 , ~1, |-assoc

)

cc Vic = 0.

the previous equation, |-norma

Thus C* is functional. Proof of (2.285): First notice that

c^ic c (c*|Di nc)|c

(2 .280)

c (c*|Di ncnvic )\c { {

2

\

>

)

rot —)

|

(2 .278)

c

(2 .248)

|-normal Consequently two cases

n (c|Di u C|D[) n C|Di) u (

fl-dist, |-dist def. of C the previous equation.

Proof of (2.286): Let R = C\C~CV. First, we have

^

^

^L/

L/ I I u

= 0IC" 1 = 0

I I JX ui j u

1

1

roil,

(2.248) -normal

1

19. PROJECTION FUNCTIONS

From this we get

u cF) n

=cn

two cases

= Cn n-dist, |-dist previous inclusion Finally,

previous inclusion

c idle"

C

is functional

= c-\ Parts of Th. 45 were designed for the proof of the following theorem. are

T h e o r e m 46. For every class C, and all x,y,z £ V, the following equivalent:

(2.287) (2.288)

statements

{x,y)eCVA{x,z)£C*, 3u3v(yw({x,w)eC

<^ w = uVw = v)A

Vw((u,w) e C e » = !/)A

IfC = E" 1 then (2.287) and (2.288) are also equivalent to (2.289)

x=

{y,z). v

PROOF. Assume that (x,y) 6 C and {x,z) £ C*. Since C^ = C*|C, there

is some i i e V such that (2.290)

(x,M)eC*

and (u,y) EC.

Now C* C C and C C C, so (2.291)

(X,M)GC

and

(«,j)6C.

Since C* = C|C U C ° | C n (C|C~ C" 7 )^, the assumption that (x, z) € C* leads to two cases. Case A. Assume O 9Q91

IT- ?\ ci O'^\r'

2. SET THEORY

Then (x,z) € C°|C, and C*|C = C*|(C n C|Di) by (2.285), so there is some » 6 V such that {x,v)€C<>

(2.293)

and (v,z) <E C n C | D i .

But C * C C, so (2.294)

(x,v)eC

and ( u , « ) € C .

by (2.292), (z,v) £ DilC" 1 by (2.293), and (a;,?;) G C

Now (x,z) £ (CIC^C^)^ by (2.294), so we have

(X,v) ec 1

Also, (y,x) e C ' " , so

By (2.286), C ^ K C n ( C | C ~ C^)""|DiIC" 1 ) C C " 1 , so (2.295)

(v,y)eC.

Next we prove Vw({v,w) £ C > w = y \/ w = z). Consider an arbitrary i u £ V. If u> = 2/ or UJ = 2, then (v,ui) £ C by (2.295) or (2.294), respectively. For t h e converse, suppose (2.296)

{v,w)€C.

By (2.294) and (2.296), (2.297)

{x,w)€C\C.

If w = y then we are done, so suppose (2.298)

{y,w)€D\.

Then (x,w) G C^Di by (2.298) and the hypothesis that (x,y) £ C 9 . We have C^IDi C V 2 ~ C ' since Cv is functional, so (a;, to) £ C^. From this and (2.297) we get (2.299) Now (a;,«) G {C^^C^y

(w,a;> € ( C | C ~ O ~ \ by (2.292), so

by (2.299). But (C\C ~ C " ) " 1 | ( C | C ~ C ^ ) ^ C Id by (2.249), so (w,«> £ Id, i.e., w = z, as desired. w = y). Consider an arbitrary w £ V. Next we prove V m ((u,w) £ C First note that if w = y then (u,w) € C by (2.291). For the converse, assume (u,w) £ C. From this assumption, (2.290), and (2.249), we get <«;,!/> G C - ' I C C Id, so w = y, as desired.

19. PROJECTION FUNCTIONS

81

Finally, we show that Vw((x, w) 6 C <S> w = uVw = v). Let u i 6 V . If w = u or w = v, then (x,w) G C by (2.291) or (2.294), respectively. For the converse, assume (2.300)

{x,w)eC.

If w = u we are done, so assume (2.301)

<M,w)GDi.

From (2.290), (2.300), and (2.301) we get (x,w) G C l~l C*|Di. ( C n C * | D i ) ^ by (2.293), so

But (x,v) G

(w,v) G ( c r n c " l > | D i ) " 1 | ( c n c r * | D i ) ^ c id, by (2.249). Thus w = v, as desired. Case B. Assume (x,z)eC\C. 1

By (2.283), C\C = (C* n C )^ = C^ n C*, so (2.302)

(1,2)6^

and (x, ^} e (C* n C)\C.

But C^ is functional and (x,y) G C^ by hypothesis, so it follows from (2.302) that y = z and (2.303)

{x,y)e(C*nC)\C.

By (2.303), there is some v G V such that (2.304)

{x,v)eC*,

{x,v)eC,

and

{v,y)eC.

However, we have (x,u) G C* by (2.290), and C* is functional, so (2.304) implies that u = v. Hence (2.305)

and (u,y) E C .

{x,u)€C

Since u = v and y = z, all we need to show is that Vw(((x,w) e C

w = M)A

({u,w) £ C <5 w = y)). Let w G V. We have (X,M) G C and (w,y) G C by (2.291), and also by (2.305). If (x,w) G C then, by (2.305), (W,M) G C'^C C Id, SO«J = IJ, and if (u,w) G C then, by (2.305), {w,y) 6 C " 1 ^ C Id, so w = j / , as desired. This completes the proof of (2.288) from (2.287). To complete the proof, we must assume there are u, v G V such that (2. /,. e . ,

V m ((x, w) € C <S> w = u V w = v), Vw({u,w)

£C

w = y),

y w ( { v , w ) e C < ^ w = y V w = z)],

2. SET THEORY

and show that (2.287) holds, i.e., (x,y) € C^ and (x,z) € C*. This is easy but tedious, and we only do part of it. Suppose u = v. Then {y} = {y, z}, hence y = z. In this case, the assumptions are equivalent to Vw((x,w) G C <=> w = u), Vw({u,w) € C O w = y), hence also equivalent to (2.305). From (2.305) and (2.283) we get {x,y) = (x,z) e C\C = C c ? n C * , as desired.

D 3

Define binary relations P, Q, and V as follows.

(2.306)

P := (E" 1 )^ n ( E " 1 ) * ^ ,

(2.307)

Q:=(E- 1 )*n(E- 1 ) ( ; '|V,

(2.308)

V3 := ( E - ^ I V n ( E " 1 ) * ^ .

The notation used to designate V2 and V3 suggests that we have two operations that may be performed on arbitrary classes and that these operations are indicated by a superscript "2" or "3", but this is not the case at this time. Such an operation will be defined later, once we have enough axioms to insure the existence, for any two classes A and B, of their direct product AxB. The exponential notation used now is simply designed to (eventually) be in agreement with notation introduced later in (2.357)-(2.360). We do already have enough axioms to define various products of V with itself, such as (V2)2 on so on. The next theorem follows immediately from (2.306)-(2.308) and Th. 46. Theorem 47. (2.309)

P=

{(x,y,x):x,yeV},

(2.310)

Q=

{(x,y,y):x,y£\/},

(2.311) (2.312) (2.313)

3

V = {(x,y,z) 2

:I,J,^V} =

1

P|V = Q|V,

1

V = p- |Q = CT |P = V|P = V|Q, Id = P | P " 1 n Q | Q " 1 ,

The keys facts about P and Q are that they are functional by (2.282), (2.284), (2.306), and (2.307), that they are pairing functions, in the sense that (2.313) holds, and they satisfy the uniqueness condition (2.313). Here are some additional equations that show what relations result from composing P and Q in all possible pairs. P|P = {(x,y,z,x)

: x,y,z € V},

P|Q = {(x, y, z, y) : x,y,z £ V } , =

{{x,{y,z),y):x,y,zeV},

=

{(x,(y,z),z):x,y,ze\/}.

A rather more interesting and useful equation involving P and Q appears in the next theorem.

19. PROJECTION FUNCTIONS

Theorem 48. For all classes A, B, X, and Y,

A\B

n X\Y = {A}?'1 n XIQ-^KPIB n Q\Y).

Tarski's QRA theorem (see Tarski [230, VII], Tarski-Givant [240, 8.4(iii)], Maddux [139, Cor. 8], Th. 427 below) says that a relation algebra is representable if it has abstract algebraic versions of P and Q. More precisely, if 21 is a relation algebra, p, q £ A, p;p < V, q;q < 1', and p;q = 1, then 21 is a representable relation algebra. QRA is the class of algebras satisfying the hypotheses of this theorem, so the theorem states that QRA C RRA. The original proof of Tarski's QRA theorem depended on Taxski-Givant [240, 4.4(xxxvii)], a theorem which asserts that the formalisms Cx and C+ axe equipollent in means of proof relative to a particulax sentence which says that quasi-projections exist; see Th. 573 below. This equipollence theorem is in turn a corollary of the Main Mapping Theorem for Cx and C+ (see Taxski-Givant [240, 4.4(xxxiv)], remarks on p. ix, and the Main Mapping Theorem (7.187) below). In fact, each of these three results can be used to give short proofs of the other two: QRA theorem, the relative equipollence theorem, and the Main Mapping Theorem for Cx and C+. The QRA theorem could be used as a starting point if it had a proof that does not use the Main Mapping Theorem for Cx and C+. Since the QRA theorem is purely algebraic, such a proof should also be purely algebraic and remain within the theory of relation algebras. Initial attempts in fall 1973 to discover such an abstract algebraic proof of the QRA theorem led to the consideration of the following test case, which is an equation that is true in every QRA. (2.314)

a;b x ; y = (a;p x;q);(p;b

q;y).

The advantage of this equation is that it is shorter and simpler than some other equations that fail in some RA and yet hold in every RRA, such as (L) and (M). It also directly involves the functional elements p and q, whose presence in the algebra is required for representability. By the end of 1973 an algebraic proof (see p. 415) of Tarski's QRA theorem provided a method for creating equational derivations of (2.314). A different proof of a stronger theorem from Maddux [139] (that tabular relation algebras are representable; see Th. 423 below), was published in Maddux [138, Th. 7]. Consequently, a short proof of the relative equipollence in proof of £ x and C+ , based on the QRA theorem, was added to Tarski-Givant [240] in 1986 after Tarski's death (see [240, fn.3*, p. 107, fn. 1*, p. 242-3, fn.3*, p. 244]). Also, in 1975 the original proof of the Main Mapping Theorem for £ x and C+ was replaced in Tarski-Givant [240, §4.4] by a similar syntactical proof that was not nearly as long (see the historical remarks in TarskiGivant [240, §4.3]). An equational derivation of (2.314) plays a key role as the starting point of this streamlined proof (see the proof of Tarski-Givant [240, Th.4.1(viii)]). That proof makes essential use of the assumption that p;q = 1. In an arbitrary relation algebra, under fewer assumptions on p and q, the equation (2.314) need not hold (see Berghammer-Zierer [18, p. 127], SchmidtStrohlein [213, p. 162], and Maddux [154]).

84

2. SET THEORY

20. Boolean and relative operations on sets In this section we show that various operations on sets (namely intersection, difference, relative multiplication, and conversion) exist, when restricted to sets, as classes of ordered pairs. These classes are used later in the construction of algebras of sets and relations, and their constructions illustrate the methods used in proving the Relation and Class Existence Theorems. T h e o r e m 49. The following

binary relations

exist and are

{((x,y)

,xUy)

: x,y

€ V},

{{{x,y}

,xCly)

: x,y

G V},

{((x,y),x~y) {((x,y),x\y)

functional:

:x,y€V}, :x,y

€ V},

Before the proof, a remark. Later we prove that the union, intersection, difference, direct product, relative product, and converses of two sets are sets. However, the complement of a set may not be a set. The class {(z,V~:c) :x£ V} is the complementation operation restricted to those sets whose complements happen to also be sets. An attempt to prove its existence might involve the following dubious equation: {(z,V~:c) : i £ V } = F o E . In general, we expect the class of all sets not belonging to a given set to be a proper class, not a set. If x £ V and V ~ x £ V then (x, V ~ x) does not exist. Intersecting a relation with V3 yields the domain-restriction of that relation to ordered pairs. (For range-restriction to ordered pairs we need the as-yet undefined V x V2; see (2.356), which is made possible by the Axiom of Singletons.) For the existence of binary relations representing union, intersection, difference, direct product, and converse of sets, it suffices to note that PROOF.

(2.315)

{«*, y),xUy}:x,ye\/}

= ((PIE"1 U Q|E x) o E) n V3,

(2.316)

{{{x, y),xny):x,yey}

= ( ( P ^ " 1 n QIE"1) o E) n V3,

(2.317)

{(( :C , 2 /),x~ 2 /): :C , 2 /GV} = ((P|E- 1 ~Q|E- 1 )oE)nV 3 ,

(2.318) {(^aT 1 ) : x 6 V} = ( E ^ ^ C T 1 n Q|P^) o E) n V3. The axioms adopted so far insure the existence of each of the relations denoted by the expressions on the right sides of these equations. These relations are functional by Th. 28. For relative product we need to use a rather more complicated construction, namely, {{{x,y),x\y):x,yeV}

=

(Ri\R2\RsoE)nV3,

where Ri = P|E- 1 |p- 1 nQ|E- 1 |CT 1 ,

88

21, RELATION EXISTENCE THEOREM

21. Relation Existence Theorem In this section we elaborate on the methods of the previous section to prove an existence theorem for every relation that can be defined by a set-bounded formula. We restrict our attention to those formulas whose variables all occur in some predetermined fixed set of 16 variables. The choice of this number of variables is merely an example. Any other small finite number of variables could be used instead, but powers of 2 are natural choices. The definitions in this section mimic the syntactic constructions of the Translation Mapping of Tarski-Givant [240, 4.4]. The Class Existence Theorem presented here (Th. 60) is essentially due to Bernays, Godel, and Tarski. This theorem have been obtained by modifying the class existence theorem of Godel [80, M2, p. 13] according to the finite axiomatizability results of Tarski-Givant [240, 6.4(iv)(v)(vi)]. Define 16 relations as follows. (2.319) (2.320)

Ro = P|P|P|P, R i = P|P|P|Q,

Rs = Q|P|P|P, R9 = Q|P|P|Q,

(2.321)

R2 = P|P|Q|P,

Rio = Q|P|Q|P,

(2.322)

Ra = P|P|Q|Q,

Rn = Q|P|Q|Q,

(2.323)

R4

(2.324)

Rs = P|Q|P|Q,

Ris = Q|Q|P|Q,

(2.325)

Re = P|Q|Q|P, RT = PIQIQIQ,

Rl4

(2.326)

= P|Q|P|P,

Rl2

RIB

= Q|Q|P|P, = Q|Q|Q|P, = QIQIQIQ

From these 16 relations we define 16 more relations Aj with 0 < * < 15 by (2.327)

A»:=

f]

R-flR-T1.

Define W to be the class of ordered pairs whose left sides are ordered pairs of ordered pairs of ordered pairs of ordered pairs: (2.328)

W : = f] R»|V. 0<s<18

By unwinding definitions we see that (2.329)

W = {<((«a, b), (c, d» , {{e, / ) , {g, ft))), a1b,c1d,e1f,g1h,i,j,k,l,m1n1o,p1q

G V}.

In accordance with the notational conventions on p. 49, the expression "a, 6, c, d, e, / , g, h,«, j , k, I, m, n, o,p,q£ V" in (2.329) is actually superfluous and could be replaced by any true statement with no change in meaning.

86

2. SET THEORY

If we refer to sets in the domain of W as 16-tuples and the image of a 16tuple under R; as its ith place whenever 0 < i < 15, then we may describe A; as the set of pairs of 16-tuples that may differ only in the i th place. The next theorem gathers together a few properties of these relations, such as, Ri is functional, Ai is an equivalence relation, and R;, Rj are a pair of projection functions whenever i + 3Theorem 50. Let 0 < i,j < 15 with i

j . Then

W = W|V,

id = R i - 1 |(R i n^) = Ri-1|Ri,

Ai n Aj = Id n W, Ai|R> C Rj.

Up to now we have proved only finitely many theorems, each explicitly stated in a discourse language that is not formally denned, called the "language of set theory" (roughly, English supplemented with mathematical symbols). Now, however, we confront the fact that it is customary to formulate and prove the Class Existence Theorem in terms of the metamathematical notions of "language" and the "collection" of all formulas in that language. For example, the "collection of set-bounded formulas", which we wish to introduce below, is not a class and, of course, it is not a set, just as formulas themselves are also not classes. These terms refer to objects in the discourse language with which we discuss and develop the theory of classes. As Godel [80] said, "[t]he general existence theorem is a metatheorem, that is, a theorem about the system, not in the system, and merely indicates once and for all, how the formal derivation would proceed in the system for any given primitive propositional function". With sufficient circumlocution it would be possible to write this section in a way that avoids formalization of the metalanguage. For simplicity of expression, however, we will proceed as if we had sufficiently formalized our metalanguage to refer to collections of formulas, symbols, variables, etc., so that we can use expressions to denote expressions. Accordingly let us say that our chosen set of 16 variables is Vie = {vo, , v 15}. One possible choice would be the set of 16 variables used in (2.329) to describe the left-hand sides of pairs in W, namely, "a", ..., "p", in which case we could correctly write, using notation formerly reserved for reference to classes and sets, that Vi6 = {"a",..., "p"}. We will generally not distinguish between expressions and expressions denoting expressions so that we can avoid the excessive use of quotation marks. By a set-bounded formula we mean any formula that can be obtained according to the following rules of construction. - Basic equality and membership statements are set-bounded if their variables are in Vi6.

21. RELATION EXISTENCE THEOREM

87

- If ip and ip a r e previously constructed set-bounded formulas and 0 < i < 15, then the following formulas are also set-bounded formulas: -up, (pAtp, (p V ip, if => if), ip <= ip, if <=> if), VVi (vi e V => if), a n d 3Vi {vt e V A j j ) .

Let *i6 be the collection of set-bounded formulas. Set-bounded formulas are known from the literature as predicative. Associate a binary relation M(i6 according to the following rules. If (p and ip are predicative formulas in 4>i6 and 0 < i, j < 15, then (2.330) (2.331) (2.332) (2.333) (2.334) (2.335) (2.336) (2.337) (2.338) (2.339)

M(vi = VJ) = {Ri n Rj)|V n W, M(vi e VJ) = (Ri|E n Rj)|v n w, M(-.
=

M(ip)\JM(iP),

M(
ip) = M(ip) ~ M((p),

M(

M(ip <^- tp) = (M(ip)'-^iM({p)) (~ M(3Vi(vi € V A ip)) = W n Aj|M M(yv.(vi e v = s - y>)) = w ~ Aj|(

This definition of M is essentially the same as the one used by Tarski-Givant [240, 4.4(vii)]. For historical remarks concerning the definition of M, see TarskiGivant [240, §4.3]. One inessential difference is that M is a computable function in Tarski-Givant [240] but is treated here as an informal set of instructions for computing an expression that denotes a desired class from a description of its members. The next two theorems are essential preliminary theorems for the Class Existence Theorem, and could be called "existence theorems for definable relations." Their proofs require neither the Axiom of Singletons nor the Class Union Axiom. (A comparable observation is made by Tarski-Givant [240, p. 182], namely, that 55 has not yet been used at a certain stage in the proof of Tarski-Givant [240, 6.4(iv)].) The first one says that M(
M[ip) C W, M{fp) = {(((((v0, «i>, (v2, v3)), «« 4 , ^5>, <«6) «7»), , V)


PROOF. The proofs of both parts proceed by induction on the complexity of set-bounded formulas. The proof of (2.340) is extremely simple. The righthand sides of (2.330)-(2.332), (2.338), and (2.339) are subclasses of W, while in

88

2. SET THEORY

(2.333)-(2.337) this conclusion follows quite readily from the appropriate inductive hypothesis. We will deal with a few of the cases in the proof of (2.341). First suppose

and (2.343)

<SM/>GW.

From (2.343) and (2.329) we conclude that there are sets vo,..., vis E V such that (2.344)

Rk(x) = vk whenever 0 < k < 15.

By (2.342) there is some z E V such that (x, z) E Ri n Rj, hence (x, z) E Ri and {x,z} £ Rj. But Rj and R; are functional, so we rewrite these last two statements using functional notation, obtaining Ri(x) = z,

(2.345)

Rj(x) = z.

From (2.344) and (2.345) we conclude that vi = Ri(x) =

Rj(x)=vj,

hence (a;, y) € {(((((v0, vi), (v2, v3)), ((v4,v5), (v6, v7))), (({.V8,Vg)

,{vio,Vn))

,((V12,V13)

,{V14,V15))))

,y)

Vi

=Vj},

as desired. For the other direction, suppose that (x, y) is in the set denoted by the expression on the right-hand side of the equation (2.341), i.e., there are sets vo,---,vi5 £ V such that (2.344) and Vi = Vj. These hypotheses imply that (x,y) E W and Ri(x) = Rj(x). Let z = Ri{x) = Rj{x). Then (x,z) (x,z) € Rj, and (z,y) £ V, so (x,y) £ (Ri l~l Rj)|V n W = M(Vi =Vj).

E Ri,

The proof of (2.331) is quite similar and almost equally simple. The proofs of (2.332)-(2.337) are quite straightforward. Most interesting are the proofs (2.338) and (2.339), which involve quantifiers. Suppose

M(V>) = {<«««0, «i> , <«2, « 3 » , ««4, V5) , (V6, V7))) , (((V8,vg)

, (Wio, Wll» , « « 1 2 , «13> , <«14, «15>>>> , V)

We wish to prove (2.347)

M(3Vi ( » , € V A V)) = (((«8, «9> , (^ 10, «ll)) , {{V12,V13} , (l)l4, « 1 5 » » , «/) : 3 ^ («i G V

Let us first prove the inclusion from left to right in (2.347). Suppose (x,y)

Ip}-

21, RELATION EXISTENCE THEOREM

89

By (2.338) this gives us (2.348)

{i,tf)eW

and (2.349)

{x,y} e Ai\M(ii).

By (2.349) there is some z G V such, that (x,z)eAi

(2.350) and (2.351)

(z,y) £ M(i>).

From (2.351) and (2.346) we conclude that there are sets vo,..., vis 6 V such that ip and (2.352)

Rfc(z) = «*. whenever 0 < & < 15.

Erom (2.350) and (2.327) we get (2.353)

(x,z) e RklRsT1 whenever i / k and 0 < & < 15.

We know from (2.348) that x is in the domain of R& whenever 0 < k < 15, so it follows from (2.352), and (2.353) that Rfc(aO = vu whenever t ^ k and 0 < fc < 15. Thus x is the same 16-tuple as z except that possibly Rt(x) ^ Ri(z), and there does exist some set Vi = Rj(z) such that ip, i.e., the conditions required for membership of (x,y) in the right-hand side of (2.347) do hold. The existential quantification occurring implicitly in the relative multiplication in (2.349) asserts that a 16-tuple exists, hence 16 sets exist, except that 15 of them are constrained to be equal to previously given sets, so the quantification is, in effect, concentrated on the existence of the set tijEV, The argument for inclusion in the opposite direction is equally elementary, and the argument for the case of universal quantification can be done similarly or reduced to the existential case. The only axioms required for the following Relation Existence Theorem are the Axioms of Extensionality, Empty Set, Complementation, Intersection, Unordered Pairs, Relative Product, Converse, and the e-Relation. Theorem 52 (Relation Existence Theorem). If ip is a set-bounded formula in # i e in which the only free variables relation {{vi,Vj) : ip} exists. In fact,

(2-354)

are Vi andvj,

0 < i,j < 15, then the

binary

{{vi^j) : tp} = RT'KRj n % ) ) .

PROOF. This theorem follows in a straightforward manner from the previous one. Suppose we have a pair ) such that (p. We may arbitrarily choose 14 more sets and construct a 16-tuple i € V whose ith and jth components are t/j and Vj. By pairing this 16-tuple with another arbitrary set we get a pair which, by Th. 50, belongs to M(tp). It is then easy to deduce, from Rj(x) = Vi and Rj(x) = Vj, that this pair belongs to Rs~x|(Rj n M(ip)). Conversely, if a pair

90

2. SET THEORY

(u, w) belongs to Ri^1|(RJ fl M((p)), then there must be some x in the domain of M(
AAVu(u£x

O u = y))).

Note that (a, a) = {{a}} for every set a 6 V, so the Axiom of Singletons implies the existence of the identity relation on any class. Theorem 53. For every class A, {(a, a) : a £ A} exists. On the basis of the preceding theorem, we may, for every class A, let A1 := {{{a}} : a e A} = {{a, a) : a € A},

(2.355)

and say that A1 is the identity relation on A, and A is the domain of A1. This is consistent with the definition of domain given on p. 66. The range of A1 is the same as its domain. The Axiom of Singletons guarantees that every class is the domain of some binary relation. It also allows us to define the direct product of the classes A and B by (2.356)

:= A^IVIB1.

AxB

Theorem 54. For any two classes A and B, Ay. B exists and (2.357)

A x B = {(x,y) : x € A Ay £ B}.

A consequence of this theorem is Axiom B5, Godel's axiom of direct product, which says that if C is a class, then V x C exists. For every class A, the direct square of A is A x A, also known as the Cartesian square of A. Let (2.358)

A2 :=AxA

(2.359)

A

(2.360)

Ai:=A3xA=

3

:= A

2

=

A1\V\A1,

xA=

(

^

1

1

1

((

and so on. According to (2.358), applied to V, V2 = V x V, but the unit relation V2 was defined earlier by V2 = V|V. However, the two definitions agree, since V x V = V|V. See the remarks following (2.308). Let R and A be arbitrary classes. R is a binary relation on A iff R C A2. The smallest binary relation on A is the empty set 0 (which qualifies as a binary relation on A because none of its elements fails to be an ordered pair with sides in A). The largest binary relation on A is A2 itself, and V2 is the largest binary relation. The diversity relation on A is A2 ~ A1. We say that R is a ternary relation on A iff R C A3, that R is a ternary relation iff R C V3, that R is a quaternary relation on A iff R C A , and that R is a quaternary relation iff R C V4. Note that the left side of an ordered pair in a ternary relation is an ordered pair.

22, AXIOM OP SINGLETONS

91

Theorem 55. (i) Every ternary relation is a binary relation. (ii) Every quaternary relation is both a ternary relation and a binary relation, (iii) / / R and S are binary (or ternary, or quaternary) relations, then so are RUS, RDS, R~S, andRe_S. (iv) If R is a binary relation then R is not a binary relation. The next theorem collects a few observations involving direct products. Theorem 56. Let A, B, C, D be classes. Then A1\B=(AxV)nB,

(2.361)

AxV = A1\V,

(2.362)

B\AX

(2.363) (2.364)

V x A = V|A\ (A x By1 =BxA,

(2.365) (2.366)

0 x ^ = 0 = ^4x0,

(2.367) (2.368)

A x B = 8 # 4 = | V B = S, ACBACCD^AxCCBxD,

(2.369) (2.370)

=Bn(VxA),

BnC^$

^> (AxB)\(C xD) = AxD,

(A x B)\(C xfl) = ( i x ( B n C))\((B f)C)x D),

(2.371)

A x (B U C) = (A x B)U (A x C),

(2.372)

Ax(BnC) = (Ax B)n(Ax C),

(2.373)

(4xB)n(Cxfl) = (inC)x(BflJ}),

(2.374)

(Ar\B)x(Cr\D) = (AxC)n(BxD),

(2.375)

ix(fl~C) = (4xB)~(ixC),

(2.376)

(AUB)xC=(AxC)U(BxC),

(2.377)

(A n B) x C = (A x C) n (B x C),

(2.378)

(i~B)xC=(ixC)~(BxC).

The Axiom of Singletons allows us an alternative route to the existence of the identity relation Id, since V1 = Id. The ability to form direct products, guaranteed by the Axiom of Singletons, provides other, somewhat simpler, ways to construct the projection functions P and Q, as shown by the following theorem. Theorem 57. : x / y} = (Di x V) n E-1|(E-1)->,

(2.379)

{{{x,y),x)

(2.380)

{«*,y> ,y>: x

(2.381)

{(a;,a;):a;eV~V 2 } = ]d~V 3 ,

(2.382)

{((m,x),x) : x G V} = (Id x V) n E ^ E " 1 ,

y} = (Di x V) n E " 1 ^ " 1 ~ E ^ K E " 1 ) ^

92

2. SET THEORY

v3n

P=

(2 .384)

-i -i Q = (Id x v)n E | E U(Di x V)nE" 1 |E" 1 '

(2 .385)

UJ

(2 .383)

E-V,

The addition of the Axiom of Singletons allows a parameterized version of the Relation Existence Theorem. Let C be a collection of letters denoting classes. The C-parameterized set-bounded formulas are defined by extending the definition of set-bounded formulas (p. 86) by including - basic statements of the form o , e C with 0 < i < 15 and C in C. Let
M(vi £ C) = Ri\{C x V) n W.

Theorem 58. If (p is a 16-variable C-parameterized set-bounded formula in *? 6 then (2.340) and (2.341) hold. PROOF. The proof is by induction on the complexity of C-parameterized setbounded formulas, and is nearly the same as the proof of Th. 51. In the proof of (2.341) there is an additional base case that arises when ip is the statement Vi E C. To prove (2.341) in one direction, assume that (x, y) E M(vi E C). By definition (2.386), we have (x,y) £ Rj|(CxV)nW. Since (x,y) £ W, there are sets vo,... ,vn E V such that Rk(x) = v^ whenever 0 < fc < 15. Since {x, y) E Ri\(C x V), there is some z £ V such that {x, z) E Ri and {z, y) E C x V. But then vi = Ri(x) = z £ C, hence

(2.387) (*, y) E {<«««o, «i), (v2, v3)), «« 4 , ^5>, <«6, *>7>» , (((us, vg), (uio, vn)), ((v12, v13), (v14,vls)))),

y) : Vi E C},

as desired. For the other direction, if (2.387) holds, then there are sets vo, , vis £ V such that Rk{x) = v^ whenever 0
D

T h e o r e m 59 (Parameterized Relation Existence Theorem). IfO < i,j < 15

and ip is a 16-variable C-parameterized set-bounded formula in i6 whose only free variables are Vi and Vj then the binary relation {{vi,Vj) : ip} exists, and (2.354) holds. 23. Class Union Axiom The class A is a union of the class B if the elements of A are the elements of elements of B. The Class Union Axiom asserts that every class A has a union: i e i & 3y(xEyAyEB)).

24. CLASS EXISTENCE THEOREM

93

By the Axiom of Extensionality, a class can have at most one union. The existence and uniqueness of a union justifies the introduction of notation for the union. For every class B, let (2.388)

\jB;={x;3v(xeyAV

The class F is a field of B if

By the Axiom of Extensionality, a class can have at most one field. Existence follows from the Axiom of Union, for if B is a binary relation then its field is ULJB. For an arbitrary class B, define its field to be the field of its relational part.

Fd(B):=\J\J(BnV2).

(2.389)

The Parameterized Relation Existence Theorem asserts the existence of any relation that is definable by a formula whose quantified variables are relativized to some class parameters. The parametric classes are represented as binary relations via the Axiom of Singletons. To prove the Class Existence Theorem we also need to be able to recover a class from some relation of which it is the domain. The Class Union Axiom does just that. In particular, the field of the identity relation A1 is just its domain, namely A:

24. Class Existence Theorem Applying the Class Union Axiom to the Parameterized Relation Existence Theorem gives the Theorem 60 (Class Existence Theorem). Suppose

{V:B,(xeCA{x,y)eR)}

exists. By making various choices for R, we get B4, B7, and B8 as special cases. Axiom B4, the Domain Axiom, asserts the existence of the class {a; : By({y,x) 6 C)}. Axiom B7 asserts the existence of {(a;, (y, z)) : (y, (z,x)) 6 C}, and Axiom B8 asserts the existence of {{a;, {y, z)) : (a;, (z, y)) € C}. For B4, let R = Q. For B7, let R = QIPIP- 1 n QIQIP-^Q" 1 n PIQ^IQ" 1 . For B8, let R = PIP" 1 n

' ^ I Q "

1

.

94

2. SET THEORY

Because of Th. 61 we may now define operators on classes that produce the domain and range of (the functional part) of a class. (2.391)

Do (R) := {o : 3b((a, b) £ R)},

(2.392)

Ra(R):={b:3a({a,b) € R)}.

Prom this point onward we will feel free to use typical mathematical constructions without necessarily giving detailed set-theoretical justifications for their existence. For example, if A and B are sets then, rather than appealing to the Class Existence Theorem for the existence of the set of functions from A to B, we simply define (2.393)

A

B := {/ : / C A x B, A1 C f\f~\

f-'\f

C B1}

and say nothing more about why this set exists.

25. Lifting relations to sets The Axiom of Replacement, stated in the next section, is closely related to, and easily expressed in terms of, an operation on relations that "lifts" a relation R between elements of a given set, producing a relation R* between subsets of that set. However, the existence of R*, and some of its properties, do not require the Axiom of Replacement. For every class R, let (2.394)

R*

^E'^RoE.

The symbol * is used for this same purpose by Henkin-Monk-Tarski [93]. The following theorem is a consequence of Th. 28. Theorem 62. For every class R, R* is a function. We say that a class Y is the ii-image of X whenever Y is the class of all sets that are right sides of pairs whose left sides are elements of X. The next theorem says that the ordered pair (X, Y) is in R* just in case X and Y are sets and the right side Y is the -R-image of the left side X. Theorem 63. For all sets X, Y £ V, {X,Y)£R*

« Y = {y: 3x{x e X A {x,y) e R)}.

If F is a functional class and (x, y) £ F, we express this in the usual manner by writing F(x) = y. Because of Th. 62, we use functional notation for R*, and write R*(X) for the image of a set X under R, i.e., R*(x) is the unique set y such that (x,x) £ R*, if such a set exists. For a given class R and a set x, there may be no set y such that (x,y) £ R*. For example, if R = V2 and 0 ^ x £ V then {«; : l ( o 6 i A (v, w) £ R)} = V, i.e., the image of any nonempty set under V2 is V, which may not be a set. Indeed, V is not a set if the Axiom of Regularity holds (see Th. 74 below) in which case (V2)* = {(0, 0)}. Suppose E is an equivalence relation and a; £ V. As was observed on p. 15, the existence of x/E, the equivalence class of x with respect to E, follows from the special case of the Class Existence Theorem expressed in (2.390). Note that E*({x}) is the equivalence class of x, and E o E, which is a function by Th. 28, maps x to its equivalence class x/E. Thus we have (2.395)

x/E = E*({x}) = (EoE)(x).

28, REPLACEMENT AXIOM

96

For any set A £ V, let A/E be the set of equivalence classes of elements of A, that is, (2.396)

A/E := {(EoE)(a) : a £ A} = (EoE)*(A),

This construction is used, for example, in the construction of quotient algebras; see (3.11). The next theorem states that the image of a set under a smaller relation R is in the subset relation to any image of that same set under a larger relation S, or, more formally, VHVSCR Q S => V4R*(x) C S*(x))).

Theorem 64. If R C S then S*" 1 ^* C E ^ f E and R* n S*|V = fl* n PROOF. First, we have

(2.394) (2.207) (2.200) C (E~ 1 t(E- 1 |S)~ 1 )|(E" 1 |5tE)

RQS, |-mon, f-mon

1

C E" f E

(2.160)

Note that R*nS*\V D iJ'nS'KE" 1 f E) by |-monotonidty. For the other direction, apply rotation to the left-hand side and use the first part:

R* ns*|v c R* n s'KS*"1!^*) c R* n (s'lE"11 E). n 26. Replacement Axiom The Replacement Axiom says that for every set s and every functional class F there is a set b, called the image of a under F, containing the right side of every ordered pair in F whose left side is an element of s, that is, ¥ s ( s £ V = yF{F is functional => 3b(b G V A V,(y G b « 3e(z £ s A (z, y) G F))))). In the context of the axioms already assumed, the next theorem is just a restatement of the Axiom of Replacement. Theorem 65. For every class R, (2.397)

R n R\Di = 0 => V2 = R* |V,

(2.398)

V2 = (E)*|V.

Theorem 66. Every subclass of a set is a set. In particular, Id n a £ V for every set s € V.

2. SET THEORY

PROOF. Let s be a set and let C be a class. First, note that = {y : x € s A (x, y) G Id n C2}

= (C1T(s). It follows that C fl s is a set because, applying the Axiom of Replacement to the set s and the functional class C 1 , we conclude that (C1)*(s) is a set. In case C is a subclass of s, i.e., C C s, we have hence C is a set. Id PI s is the intersection of the class Id with the set s, and hence is a set. Theorem 67. For any classes R and S, R*\S* C (i^S*)*. If R is functional, then R*\S* = (R\S)*. If R and S are sets then R*\S* = (R\S)*. PROOF. The first part requires only the calculus of relations: R*\S* = (E" 1 |floE)|(E" 1 |S'oE)

(2.394)

CE^I^ISoE

Th. 2.221

= (R\S)*

|-assoc, (2.394)

Equality holds when R is functional, but requires the Axiom of Replacement. Assume R is functional. By the first part, we need only show (i^S1)* C i?*|5*. First, derive the inclusion R"-1\{R\Sy

= (E- 1 | J RoE)~ 1 |(E- 1 |.R|SoE)

Th. 2.394

= (E- 1 o(E- 1 |iZ)- 1 )|(E- 1 |iZ|5oE)

(2.221)

1

CE~ |S'oE

Th. 25

= 5"

Th. 2.394

and use it in this calculation:

(R\sy = (R\sy nv 2 = (R\Sy n iT|V

R is functional, (2.397)

cirKvnir^K-RlS)*) = R*\(R*-1\(R\Sy)

rot

C iJ'IS*

previous inclusion.

27. Set Union Axiom A class B is a set union of the class A if B is a set and B is the union of A, that is, U A = B G V. The Set Union Axiom asserts that the union of every set is a set: Vs(s e v = > 3u(u e V A V ^ I e u <s> 3^(2; ey Ay e s)))).

27, SET UNION AXIOM

97

Every class has a union, so the Set Union Axiom is equivalent to Vj,(fl G V => U s g V). However, for any sets u, a 6 V, we have \Js = u « {s,u) e E ^ l E ^ o E , so we may use the notation (J not only as an operation symbol on classes, but also as a class in its own right. Therefore, let

U:=E"1IE"1<>E=«E"1)*)"1'

(2.399) and, similarly, let

Q:=(E=TtE"1)oE.

(2.400)

The Set Union Axiom is equivalent to the equation V2 = V| [j. Theorem 68. Ifx,y,R€V

then xUy,xf)y,x~y

e V and R* \V = V2.

PROOF. By the Axioms of Extensionality and Unordered Pairs, {x,y} is a set since x and y are sets. By the Set Union Axiom, IJi^iJ/} ' s a se ^- But Uixiy} = x\Jy,so xUyis a, set. We also get xny G V and x~y E V because, by Th. 66, subclasses of sets are sets. By Th. 60, the class {b : 3 o (a € x A (a, b) 6 R)} exists. Furthermore,

R*(x) = {6 : 3a(a G x A (a,b) e R)} C {J{JR. If R is a set, then so is [J {J R by the Set Union Axiom, hence R* (a;) is a set since subclasses of sets are sets. Define the successor of a set 3 G V to be a U {a}. By the previous theorem, the successor of a set is a set. Note that for all a, b G V, b is the successor of a iff (a, b) € (E" 1 U Id) o E. By starting with the empty set and forming successors, we get natural numbers. The first few natural numbers are defined thus: O:=0,

l:=0U{0},

2:=1U{1},

3 := 2 U {2},

4 := 3 U {3},

It follows that 0 =0

i = {o} = m, 4 = {0,1,2,3} = {0, {0}, {0, {0}}, {0, {0}, {0, {0}}}},

The existence of the set of all natural numbers, namely w, must wait for the Axiom of Infinity. The next theorem tells us that R* is additive and normal, that is, R*(xUy) =R*(z)UR*(y) and R*($) = 0 for any x,y G V. Theorem 69. Assume R is any class, (x,a) G R*, and (t/,6) G R*. Then (xUy,aUb}eR\ Furthermore, (0,0) g R*, so R*(0) = %.

98

2. SET THEORY

PROOF. By the hypotheses, x, y, a, 6 are sets. By Th. 68, x U y and a U 6 are also sets, hence (x U y, a U 6) € V.

28. Powerset Axiom A class B is a powerset of a class A if B is a set, and the elements of B are the subsets of A, i.e., Vx{x £ B <=> x C A A x is a set). The Powerset Axiom asserts that every set A has a powerset: Vyi(^4 € V => 3 s ( B is a powerset of ^4)). The uniqueness of powersets follows from the Axiom of Extensionality, so for any set A 6 V, we may define 56 (A) to be the powerset of A: (2.401)

Sb(A):={x:x€V

A x CA}.

The Powerset Axiom is equivalent to

yA(A e v => Sb(A)e V), and is also equivalent to the following equation in the calculus of relations.

Alternatively, we may simply let (2.402)

SJ^tE-'fEJoE.

Then 56 is a functional class by Th. 28, so we may use the functional notation "56 (A)" and regard (2.401) as a theorem instead of a definition. Finally, we may express the Powerset Axiom by this equation.

V2 = Sb\V. Theorem 70. Ifx,yeV

then x\y,x x y, x^1 e V.

PROOF. By Th. 68, x U y is a set since x and y are sets. By the Powerset Axiom, applied twice, 56 (56 (x U y)) is a set. However, xxy C 56 (56 {x U y)), so x x y is a set because every subclass of a set is a set. By the Union Axiom, applied twice, 56 (56 (x U y)) is a set, hence (56 (56 (x U y))) x (56 (56 (x U y))) is also a set by what we have proved so far. However, ( I X J ) U (^ - 1 ) ^ (56 (56 (x U y))) x (56 (56 (x U J/))), so x\y and x~x are sets. 29. Partial orderings, meets, joins, and lattices A class R is domain-reflexive iff R = (Id PI R)\R, and range-reflexive iff R = i?|(ld fl R). By (2.103), every domain-reflexive (and every range-reflexive) class is a binary relation. Furthermore, by (2.191), (2.187), and (2.140), R is domain-reflexive iff R = (Id l~l R'1)^ and range-reflexive iff R = R\(\d n -R"1). R is reflexive iff R is both domain-reflexive and range-reflexive, that is, R = (Id D _R)|.R|(ld D R). An alternative form of this definition, used by Tarski-Givant [225, p. 110], is that R is reflexive iff R D Id n (R\V U V|.R). Note that R is reflexive iff \/x\/y({x, y) € R V {y, x) € R => {x, x) € R).

29. PARTIAL ORDERINGS, MEETS, JOINS, AND LATTICES

99

R is antisymmetric iff R fl iZ" 1 C Id, hence R is antisymmetric iff VxVy({x,y) £ R A {y,x) £ R => x = y). R is asymmetric iff R fl iZ"1 = 0, hence R is asymmetric iff \/x\/y({x,y)

£ R => (y,x) <£ R).

1

R is linear iff iZ|V|iZ C R U iZ" U Id. R is a preorder iff R is reflexive and transitive. R is a partial ordering iff R is reflexive, transitive, and antisymmetric. If R is a partial ordering then R is a binary relation and Ra (R) = Do (R). R is a chain iff R is a linear partial ordering. R is a partial ordering of a class X iff R is a partial ordering and X = Do (R) = Ra (R). If x is in the field of R and there is no y such that x ^ y and (x, y) £ iZ, then x is said to be a maximal element for R (or R-maximal element). Similarly, if x is in the field of R and there is no y such that x ^ y and (y, x) E iZ, then x is a minimal element for R (or iZ-minimal element). A class X is well-ordered by R iff R is a partial ordering of X and every nonempty subset of X has an iZ-minimal element. If (y, x) £ R for every y £ X, then a; is an upper iZ-bound of X. If x is an upper iZ-bound of X and (x, z) E R for every upper iZ-bound 2 of X, then x is a least upper iZ-bound of X. If (x, y) E R for every y E X, then a; is a lower iZ-bound of X. If x is a lower iZ-bound of X and (a, x) € R for every lower iZ-bound ,z of X, then x is a greatest lower iZ-bound of X. If X is empty, then every class is an upper iZ-bound of X. On the other hand, if X is not empty, then no proper class can be an upper iZ-bound of X, for if x E X and u is an upper iZ-bound of X, then {x, u) E R, which implies that u is a set in the range of R. If x and y are both least upper iZ-bounds of X ^ 0, then (x, y) £ R and (3/, x) £ R. Consequently, if R is antisymmetric, then every nonempty class X has at most one least upper iZ-bound, denoted, if it exists, by ^2R X. We express the non-existence of a least upper iZ-bound of X by saying " 5 ^ B X does not exist". Similarly, if R is antisymmetric then a nonempty class X can have at most one greatest lower iZ-bound, denoted by n * X if it exists, and if a nonempty class X has no greatest lower iZ-bound we say U]JR X does not exist". These notions are most appropriate when applied to a partial ordering. In the context of a fixed partial ordering R, we may write simply ^ and n instead of ^ f l and J ^ In the next theorem we formulate some of these definitions in the calculus of relations. Theorem 71. Suppose X, u, and U are sets and R is any class. Then (2.403)

u is an upper R-bound of X « (X,u) E E^fiZ & X

(2.404)

x{u}CR,

u is a lower R-bound of X « (X,u) E E^f-R" 1 « W x X C i Z ,

(2.405)

U is the set of upper R-bounds of X

2. SET THEORY

(2.406)

U is the set of lower R-bounds of X

« (x,u) e (Ei:rt-R"1)oE, (2.407)

u is an upper R-bound of the set of lower R-bounds of X

(2.408)

« {x,u) e E^liFiffl, u is a lower R-bound of the set of upper R-bounds of X « (X,u) £

E-^RjR-1,

(2.409)

u is a greatest lower R-bound of X

(2.410)

u is a least upper R-bound of X «

(X,«) € (E

iZ is called a lattice ordering of B iff iZ is a partial ordering of B and, for every two-element subset X C B, ^X and n ^ exist. A lattice ordering of B is bounded if the least upper bound ^2 B a n d the greatest lower bound Y\B both exist. A lattice ordering is complete if ~^X and n ^ exist for every subset X C B. A lattice ordering of B is distributive if ]J{x, ^2{y,z}} = 12{n.{x>y}>Tl{x>z}} f° ra n x,y,z € B. A lattice ordering of B is complemented if R is bounded and, for every x € B, there is some y such that Y\{xi y} = I I B and $^{x; v) = S ^ - ^ lattice ordering is Boolean if it is distributive and complemented. In a distributive lattice, an element can have at most one complement. The converse was disproved by R. P. Dilworth [68], who showed there are non-distributive lattices in which every element has a unique complement.

30. Axiom of Infinity The Axiom of Infinity is that an infinite set exists. There are many ways to say this. In the version used by Godel [80], the Axiom of Infinity states that there is a nonempty set C such that every element of C is a proper subset of another element of C: 3c{Mx

e C E V ) A Vx(x e C = * 3y(x CyAye

C))).

Godel's axiom of infinity can also be expressed as an equation in the calculus of relations, such as (v|E~V|((E-i|E~E-i|E)|EnE))|E|V. Here is another axiom of infinity, which asserts the existence of a set containing the empty set and closed under the formation of successors: 3o(0 e C e V A V ^ e C ^ i u W e C)).

31. AXIOM OF CHOICE

101

With the help of this axiom we may deduce the existence of the set of natural numbers, (2.411)

u:= {0,1,2,3,4,5,6,...}. 31. Axiom of Choice

The strong form of the Axiom of Choice is that there is universal choice function, i.e., a functional binary relation that picks an element of every nonempty set: 3c (C is functional A Vx(x is a nonempty set <=> 3y(y E x A (x, y) 6 C))). Let C be a class whose existence is guaranteed by the Axiom of Choice. Recall, for comparison, that the Axiom of the e-Relation asserts the existence of the class E = {(x, y) : x € y}. Using E, the conditions satisfied by C can be expressed more succinctly in the calculus of relations. Here is one way: CCE"1,

CnC|Di=0,

C|V = E"1|V.

The Axiom of Choice is equivalent to the assertion that every binary relation contains a function with the same domain. Here are two well-known consequences of the Axiom of Choice. Theorem 72 (The Well-Ordering Theorem, Zermelo [258, 259]). Every set can be well ordered. PROOF. Let s be any set. Prom the Axiom of Choice first deduce the existence of a function g : Sb (s) ~{s} —> s such that g(x) £ s ~ x for every proper subset x C s. To obtain g, just compose complementation relative to s (see (2.317)) with the universal choice function:

(2.412)

g := ( ^ ( ( ( P l i ^ s)}^1)

~ Q ^ " 1 o E) n V3)|C.

Let K C Sb (s) be the intersection of all sets X satisfying the following conditions: x £ X => x U {g(x)} £ X,

Y

a

^-{JY

ex.

K exists by the Class Existence Theorem and the form of its definition by a set-bounded formula: (2.413)

K := {k : Vx(X E V A (V^a; E X => x U {g(x)} E X)

feex)}. Then s = \J K and s is well-ordered by R, the relation that holds between a e s and b G s iff there is some k e K such that a G k and b £ k, that is, (2.414)

R:=

102

2. SET THEORY

Theorem 73 (Zorn's Lemma [261]). If R is a partial ordering of X and every linearly ordered subset of X has an upper R-bound, then there is a maximal element for R. 32. Axiom of Regularity The Axiom of Regularity states that every nonempty class has an element with which it has no common elements: VA(3b(b G A) => 36(6 e i A V ^ ^ V i ^ 6))). This axiom is essentially due to J. von Neumann [253, p. 231, Axiom VI4]; the version used here is due to P. Bernays; see [80, p. 7]. It was proved consistent relative to the other axioms by von Neumann [253]. As Godel [80, p. 6] said, it "is not indispensable, but it simplifies considerably the later work". Using it, we can now prove that the class of all sets is not a set. Theorem 74. For every class X, X 0 X. In particular, V ^ V. PROOF. Suppose X e X. Then X G V, hence {X} £ V. Every element of {X} has an element in common with X, namely X itself, contradicting the Axiom of Regularity.

33. Ordinals and cardinals A class X is an ordinal if every element of X is a subset of X and X is wellordered by E. This definition is due to J. von Neumann [253]. R. M. Robinson [211] noted that, in the definition of ordinal, the condition that X be well-ordered by E can be replaced, owing to the Axiom of Regularity, by the condition that E is linear on X. A class X is a cardinal if X is an ordinal and there is no bijection between X and some element of X. The Axiom of Choice is a major ingredient in the proof that for every set X 6 V there is a unique cardinal \X\, called the cardinality of X, for which there exists a bijection with domain X and range |X|. For more of the elementary theory of ordinals and cardinals, and more complete developments of basic set theory, see, for example, Godel [80], Kelley [122], Mendelson [169], or texts by Vaught [251], Suppes [223], Halmos [88], Monk [179]. We will simply assume elementary results about ordinals, cardinals, real numbers, complex numbers, integers, number theory, and so on. 34. Dedekind-MacNeille completion Dedekind's method of cuts, as generalized to partial orderings by MacNeille [136], is presented here in details using the calculus of relations. There is a precedent for this. Long before MacNeille's work, Schroder [215, §23, pp. 346387] used the calculus of relations to study "Dedekind's Kettentheorie". The Dedekind-MacNeille completion provides an alternative method for constructing the completion of a Boolean algebra; see Th. 215 and the proof of Th. 217.

34. DEDEKIND-MacNeille COMPLETION

103

34.1. Upper and lower bound operators. We start with upper and lower bound operators. For every class R, (2.415)

R1':= (E^1-\ R) o E,

(2.416)

Rl : = (E I T t-R~ 1 )< > E -

By Th. 28, R^ and R^ are functional. An ordered pair of sets (x, y) is in R} if the right side y is the set of upper _R-bounds of the left side x: (2.417)

(x,y) £Rr <S> y = {b : Va(a G x => (a,b) G R)} G V.

If x = 0 then {b : Va(o G x => (a, 6) G R)} = V, and V is not a set by Th. 74, so 0 is never in the domain of R^, even when R is a set. From definitions (2.400) and (2.416) we have f| = El. Recall that the relation Id o E maps each set to its singleton, as is illustrated here: Id oE , Furthermore, if R 6 V then R o E is the function that maps each set a E V to {x : (a,x) G R}. It follows from these observations and (2.421) and (2.422) in the next theorem that

Ro E(a) = R*({a}) = R1({a}) for every set a 6 V. Other parts of the next theorem are improvements on parts of Th. 25 that result from considering E instead of an arbitrary class. Compare, for example, (2.425) with (2.227), (2.426) with (2.223), and (2.427) with (2.225). Theorem 75. Assume that S and T are arbitrary classes and R is a set. f O A ~\ Q\

( E? — \

yZi.^LO)

yTt

)

1

Z?J* — ix

,

( E? — 1 \4yTt

)

T

(2.419)

E- |V = i? |V = ^ | V ,

(2.420)

V2 = (i?oE)|V,

(2.421) (2.422)

RoE = {\doE)\R\ RoE = (Id oE)|i?*,

(2.423)

(i?" 1 )oE = (Id o E ) | ^ ,

(2.424) (2.425) (2.426)

(R

(2.427)

(R-

(2.428) (2.429)

(2.430)

(RoE)\(E-1\SoT) = R\SoT,

(floE)l(E" 1 )*

=R\E~1oE,

(2.431)

( J RoE)|E 3T = B n V 2 ,

(2.432)

(RoE^E'1 = R,

nT — ix

,

104

2. SET THEORY

(RoE)*\f]

(2.433)

= R1',

(R'1 oE)*|P| = Rl,

(2.434) (2.435)

(RoEy\{J

= R*.

R?\R^ and R^\R^ are expanding: (2.436)

R^R1 C E ^ f E ,

(2.437)

R^R* C E ^ f E -

R^ and R^ reverse inclusions: (2.438) (2.439) (2.440) (2.441)

i? t ~ 1 |(E rr tE)|.R T C E ' ^ E , ( E ^ f E) l~l E-1|V C fl^E"1 f E ) ! ^ " 1 , R^^iE^1 (E I I T tE)n(E- 1 |V)

CR^iE'1

R?\R^ and R^\R? are transitive relations, because (2.442)

i? T |i? i |i? T C JRt,

(2.443)

R^R^R1

C R1.

If R is transitive, then (2.444)

((ETTt-R"1)n(E"1|lrT

(2.445)

Ul i Z t = (^ T

PROOF. Proof of (2.419): By (2.417), to prove the inclusion E - 1 |V C i?T|V, we must show for every nonempty set x that jd : Va(a 6 i => {a,b} 6 R)} is also a set. By Th. 60, the class { 6 : V o ( a £ i => (a, b) £ R)} exists for any classes x and R, so let y = {b : Vo(a £ x => (a,b) £ R)}- Our assumption that x is nonempty implies y C R*(x). (This inclusion may fail if a; = 0.) Since R is a set, R*(x) E V by Th. 68. It follows that y is a set since subclasses of sets are sets. For the opposite direction, assume that x is the left side of a pair in .R^V. By Th. 60, {6 : Va(a Ex => (a,b) E R)} is a set, hence, as we observed after (2.417), x cannot be empty, and is therefore the left side of a pair in E ^ V . Proof of (2.420) and (2.421): By (2.223) we have (2.446)

(Id oE)|_RT = (Id o E)|((E T T t R) <> E) C (Id"f R) o E = Ro E.

Then V2 = (ldoE)|V

(2.245)

C (ldoE)|(Vn(ldoE)-1|V) -i I

= (Id oE)|(E" old)|V

rot

34. DEDEKIND-MacNeille COMPLETION

c (Id oE)|(E

tDi)|V

= (Id oE)|E" 1 = (Id oE)|i? 'IV c (R oE)|V

(2.419) (2.446)

c V2 so we have shown somewhat more than (2.420), in fact, V2 = (ldoE)|B t |V = (i?oE)|V.

(2.447)

Now Id o E, R?, and Ro E are functional by Th. 28, so (Id o E ) ^ is functional by Th. 30, hence (Id o E ) ^ = Ro E by (2.446), (2.447), and Th. 33. Proof of (2.422): By (2.221),

(2.448)

(ldoE)|iT = (ldoE)|(E" 1 |BoE) C Id \Ro E = Ro E.

By Th. 2.245, V2 = (Id o E)|V, but we also have V2 = i T | V by Th. 68 since R is a set, hence V2 = (Id o E)|iT|V. Furthermore, (Id o E)|iT|V C (Ro E)|V C V2 by (2.448), so

(2.449) V2 = (ldoE)|ir|V = CRoE)|V. By Th. 28, Id o E and RoE are functional, while R* is a function by Th. 62, so (ldoE)|iT is functional by Th. 30. Finally, (ldoE)|iT = .RoE by (2.448), (2.449), and Th. 33. Proof of (2.423) and (2.424): By (2.418), (2.421), and (2.422). Proof of (2.425): (-RoE)l(E -LoS) CRoS = (RoS) n(iioE)|v

(2.227) (2.420)

C(iJoE)|((E" 1 ofl- 1 )|(iioS'))

rot, (2.207)

C(floE)|

(2.227)

o5)

Proof of (2.426): r 0R,^S)o j r = ((E:f5)oT)r l(ii oE)|V C (R oEJKCE"1 oR- -^K^t^oT)) C (R oE)|((E~T oT) J

C (R tS)oT

(2.420) rot, (2.207) (2.223) (2.223)

Proof of (2.427): Use (2 .225). Proof of (2.428): Set 5 := E" 1 andT == E in (2.426). Proof of (2.429): R\SoT = (R\i>oT)n(R oE)|V C (Ro

1

C (Ro CR\SoT

1

l

(2.420)

*>R~ )\(R\SoT))

rot, (2.207)

^ oT)I

(2.221) (2.221)

2. SET THEORY

Proof of (2.430): Set S = E" 1 and T = E in (2.429). Proof of (2.431): (i?*E)|ErTClnV2

(2.217)

= Sn(iZoE)|V

(2.420)

1

1

C(RoE)\((E~ oR~ )\R)

rot, (2.207)

C^oE)!!^

(2.216)

Proof of (2.432): (RoE^E'1 CR

(2.214)

= RD(RoE)\\/

(2.420)

1

1

C(RoE)\((E~ oR~ )\R) CfiloEllE"

rot, (2.207)

1

(2.214)

Proof of (2.433): (i?oE)*|P) = (E- 1 |(i?*E)oE)|((E rT tE" 1 )oE)

(2.400)

= (E- 1 |(i?*E)tE- 1 )oE

(2.426)

) oE

duality, f-assoc (2.431)

= R?

(2.415)

Proof of (2.434): By (2.433) and (2.418) Proof of (2.435): (i?oE)*||J = (E" 1 |(floE)oE)|(E" 1 |E" 1 oE) 1

(2.394), (2.399)

1

= E" |(floE)|E" oE

(2.429)

1

= E" |i?oE

(2.432)

= R*

(2.394)

Proof of (2.436): Before giving the proof, note that the hypothesis (x,y) € could be written more conventionally as y = R^(R?(x)), in notation whose use is justified by the functionality of these relations. In particular, (x, R^(R^(x))} € it^l-R^, so, by the proof we are about to give, (a;, R^(R^(x))} € E" 1 f E, which, written more conventionally, becomes x C R^(R^(x)), showing that R?\R^ is expanding.

C(E- 1 t-RtE)|(E-itiZ-ifE) _

= 1^1

E

_

1

(2.200), (2.415), (2.416) ^ (2.138)

(2.162)

34. DEDEKIND-MacNeille COMPLETION

Proof of (2.438): We have

(2 .416) (2 .200), f-assoc (2 .162), f-assoc

C E-i

and (2.415) (2 .207) (2 .200)

(2.160)

C

Proof of (2.439):

rot, Peirce's Law (2.419) rot (2.438) Proof of (2.440): Apply (2.438) to R'1. Proof of (2.442):

(2.436), (2.437), |-assoc

C

(2.200), (2.416) (2.162) (2.415), (2.200) Proof of (2.444): Assume R is transitive. Then

((I^t^- 1 ) n (E-^iF |-mon Peirce's Law, f-mon, (2.137)

2. SET THEORY

R is transitive, '-moil, f-mon

C E-i

and

-mon (2.162) duality

(2.200) (2.416)

Proof of (2.445): Wee have ( E - ^ E - 1 o E)|((E-i tfl)oE)

(2.399), (2.415)

(E-ilE-if^oE

(2.426) duality (2.415)

and x

(E-lrf.ElK^t E- )oE)

(2.400), (2.394)

(E- 1 |BtfE" 1 )<>E

(2.426)

1

rT

(E- |((E f-R)oE) fE-^oE

(2.415)

E-iKCE-1 f-R)oE)|[""

duality

Ot

(2.431) duality /p— 1 4- D V

(2.415)

34.2. Restricted upper and lower bound operators. Next we define some restricted versions of the upper and lower bound operators. For every class R, (2.450)

R* :=((ErT:iiR)nV\R)oE,

(2.451)

R^ := ((E^f-R^nVli^oE.

34. DEDEKIND-MacNeille COMPLETION

109

Note that R® and R1^ are functional by Th. 28. Since R^ = (R^1)^, we state or prove some things only for R^. An ordered pair of sets (X, Y) is in R^ if the right side Y is the set of those upper _R-bounds of the left side X that are in the range of R. Indeed, for all sets X and Y we have (2.452)

( X , Y)eR1r

^ Y = {b: 3a((a, b) E R), V o (a E X => (a, b) E R)}.

If X = 0 then {6 : 3 a ((a,6> G R), V a (a G X => {a,b} G R)} = {b:3a((a,b)

ER)} = Ra(R).

It follows that if Ra (R) is a proper class then 0 is not in the domain of R^. If R is set then Ra(R) and Do (R) are also sets and (®,Ra(R)) E R*, i.e., ^ ( 0 ) = Ra(R). For similar reasons, i^(0) = Do(R). Let X £ V. If X ~ Do (R) ^ 0 then {6 : 3o((a,6> G R), Va(a G X => (a,b) G i?)} = 0 and (X, 0) G iJ*. So 0 only if X C Do (i?). Theorem 76.. Assume R is a set. Then 1 (2.453) R* =^(R- )*, Ri' = (R-1)\ (2.454)

R*: V -> Sb (Ra (R)),

(2.455)

R*: V -> Sb (Do (R)),

(2.456)

V2 == ^ | V = ^ | V ,

(2.457)

R1t == (E- f-Rf(E U R-^V)) n (E- f iZf E) n (V|i?f E)

(2.458)

R^ == (E- 1 f R-1 f(E u ij|v)) n (E- 1 f R'11E)

(2.459)

1 1 R*' ^(E- f E)|i?^ C E~ fE,

(2.460) (2.461)

tEC^KE^fE)!^"1, R"\l C E-1f(EUi?|V),

(2.462)

R"-\I ^

(2.463)

(2.466)

n (Vli?- 1 1E).

r-_ 2

C E-it(EUi?-!|V), R*\l I^IR^ C^KE^fE), ^IR1'

(2.464) (2.465)

1

1

R*\l X R*\l

Lft

C R^KE-1 f E ) , -ft

^

->Jj- 1 Dft 1 p-li

.ft

.ft , DiT I DIL

PROOF. Proof of (2.454)-(2.456): Ra (R) is a set since R is a set. For an arbitrary set X 6 V let Y = {b : 3a((a,b) E R), Va(a 6 X => (a,b) E R)}. Y exists by Th. 60, and Y is a set because it is a subclass of the set Ra (R), so (X, Y) G R1'. This shows that V2 = _R^|V. The functionality of R?, mentioned above, and the observation that the right side of every pair in _R^ is a subset of Ra (R) are enough to establish that R® : V -> Sb (Ra (R)). The claim about R1^ holds for similar

2. SET THEORY

Proof of (2.457): First, we get (E-1f.R)nV|.RtE = (E-1t-RUV|i?)fE

(2.46)

= E-1t-Rt(EUi?-1|V),

(2.146) (2.450)

!} R)nV\R)oE = ((E- 1 ti?)nv|BtE)n(((E- 1 t-R)nv|i?)fE)

(2.200)

= (E- 1 ti?t(EUB- 1 |V))n(E- 1 t-RtE)n(V|i?tE) Proof of (2.458): Apply (2.457) to i T 1 . Proof of (2.459): We have

(2.119)

(2.457) (2.162), f-assoc. and

c V|(V|i?tE) c V|flfE

|-mon Peirce's Law, |-assoc

so

(2.467)

( E-T t E)|^ CE^f^tE n V|i?fE

We then get (2.459) as follows R« ' K E - i f E 1 C(((E=Tffl) nv|i?)oE)~~ \((E^^R)nV\R)^E)

(2.450), (2.467)

= (E-1o((E~~]R)nv\Rr^KaE-Tf^nvi^tE)

(2.207)

CiE-'Ui^ C E^tE

Tj

tR)nv\R] f1)\(((E^^R)nV\R)^E)

(2.200) (2.160)

It may be worth noting that we did not need to assume that R is a set. That assumption is used in the next proof, however. Proof of (2.460): (2.456) rot, |-assoc (2.459)

34. DEDEKIND-MacNeille COMPLETION

Proof of (2.461): (2.457), (2.458)

= (E" 1 f(i?f E))|((i?f

f-assoc, "Maws (2.162)

Proof of (2.463): First, we have (2.458) "Maws Peirce's Law

C E T

= E~ fi?|V

|-assoc, (2.102), (2.106)

so (2.461) rot J;

rT

rT

= B |((E t(Eu!RlV))n(E t^|V))

inclusion above

Th. 12 f E)

C

, |-mon

Proof of (2.464): Substitute R'1 for R in (2.463). Proof of (2.465): By (2.464), we have C

E).

Furthermore,

(2.174), |-assoc C

(2.463), |-assoc

C

R^ is functional

C R^IR^KE'1 fE)

(2.459)

1

Now R ^ and R® are functional, so R*^ is also functional by Th. 30. Then R1^^ distributes over intersection by (2.264). Combining these observations with the previous two equations, we obtain

C

f E) n

t E)

112

2.

SET THEORY

= iZ^|iZ^|((E^tE)n(E- 1 tE))

(2.264) (2.178)

Let X := iZ^|iZ^. Then we have shown X|X C X , so we also have

x = xnv|x-1 c x n (y n x\x)\x-' = xnx|x|x~ 1

(2.456) rot

CXIXKX-^X- 1 !^- 1 !

rot

X\X C X, -mon X is functional

(— Y1 /1 J\"V— 1 1 \ jV1 \ LA. LA. Y\)

CXiXlId

= x\x as desired. Proof of (2.466): By (2.453) and (2.465). 34.3. Cones. For every a; £ V, (R~1)* ({x}) is called the ii-cone of a;. Note that y € (R'1)* ({%}) <=> (y,x) € R, and R~1oE sends elements to their ii-cones: R-1 oE = {{x^R-1)*

({*})> : x € V}.

If iZ is transitive then, according to (2.220) with E = E, the function iZ^1 o E sends pairs related by R to pairs related by inclusion, that is, for all p , g € V , (R-'o E)(p) C (fl- 1 O E)(q).

{p,q)£R^

Theorem 77. Assume R is a set. Then (2.468)

(

(2.469)

(ld

PROOF.

l

d

1

^

We have

i?|(ld o E)|i^ C (ijf E ) | ^

Peirce's Law

C (BtE)|(E- 1 t-R- 1 t(EUiZ|V)) j

(2.200), (2.458) (2.153)

From these two equations it follows by Th. 12 that i?|(ld o E)|iZ^ C E, which in turn implies (Id o E)|iZ^ C R~x j- E by (2.164). Combining this last equation with

= (DifE)|ii* "fi?

J

tE)

(2.458)

34. DEDEKIND-MacNeille COMPLETION

Ci^fE

t-assoc,(2.153)

yields (2.470)

(ldoE)|#

CR^oE. 2

For the opposite inclusion, recall that V = (Id o E)|i2*|V by (2.245) and (2.456). But then we also have V2 = (Id o E)|.R*|V C (R-1 o E)|V C V 2 , by (2.470), hence (Id o E)|ii*|V = (R'1 o E)|V. We get (Id o E)\R? = R'1 o E by

Th. 33. Applying this result to R~x gives (Id o E)|i^ = Ro E. Theorem 78. Assume ii G V and R is a transitive relation. Then R-1 o E preserves greatest lower R-bounda in the sense that ifu is a greatest lower R-bound ofX ^ 0j then the R-cone ofu is the intersection of the set of R-cones of elements

ofX. PROOF. Assume n is a greatest lower .R-bound of X ^ 0. We can express this assumption in several ways. We could write u = JJ X if R were antisymmetric, for in that case greatest lower bounds are unique, Yl is functional, and the desired conclusion could be expressed as follows:

u = Y,*X => (R-1 oE)(u) =f]((R-1 oET(X)). In the absence of antisymmetry, we are assuming, by (2.409), {x,u) e (E^jR-1)

n ( E - ' l i F i f-R) n E- X |V.

Let C be the ii-cone of u, that is, C = (R'1 o E)(«) or {«, C) e R'1 o E. By (2.444) and (2.434), we have ((E^jR-1)

n (E" 1 !!^t-R^K-R" 1 o E) C R1 = ( i r 1 o E)* P

so {X, C) € (R~x o E)*| f|, which, written differently, asserts that C = E)*(X)), as desired. A more conventional and less computational proof runs as follows. Suppose that w is in the iZ-cone of w. Since w is a lower bound of X, this gives us (w, u), («, a;} € R for every a; € X. But R is transitive, so {w, x) 6 R for every x 6 X, hence w is a lower .R-bound of X. Therefore x is an element of every JJ-cone of every element of X, i.e., u is an element of the intersection of the il-cones of the elements of X. Thus transitivity of R implies

For the other direction, assume w is in the intersection of the .R-cones of elements of X, Then w is in the .R-cone of every element of X, i.e., w is a lower .R-bound of X. But u is a greatest lower .R-bound of X, so {w,u) £ R, hence w is in the .R-cone of u. Thus (This inclusion holds for all relations, without assuming transitivity.)

114

2. SET THEORY

The next theorem shows that R^1 o E converts least upper iZ-bounds into (-R^I-R^)-closures of unions of cones whenever R is transitive and domain-reflexive. When R is also antisymmetric, this can be expressed as follows:

OR"1 o E)(J2RX) = i ^ f l j K f l " 1 o E)* (X)))). Theorem 79. Assume R 6 V and R is a domain-reflexive and transitive relation. If u is a least upper R-bound of X ^ 0, then the R-cone of u is the set of lower bounds of the set of upper bounds of the union of the set of R-cones of elements of X. PROOF. R is transitive and domain-reflexive, so R\R C R and (Id C\R)\R = R. Recall from (2.410) that u is a least upper .R-bound of X iff (X,u) € J where J := ( E ^ f f l ) n (E-^RjR-1)

(2.471)

nE ^ V .

1

Let C be the .R-cone of u, that is, C = (R' o E)(M) or (u, C) £ R'1 o E. Then {X,C)eJ\{R~1oE).

(2.472)

We wish to show that (X,C) G (R'1 o E)*|U|^|fi 4 , which is equivalent to C = (R^lR^dJdR-1 o E)*(X))). But (R-1 o E)*| \J = (R'1)* by (2.435), so we need only show (X,C) £ (R-1y\Rit\R>;>. Let (2.473)

A :=

(E-loR)\,rl\(R-l)*\R1t\Ris-.

We will show that E\A C E and ^4|E—1 C E" 1 . From these two inclusions we conclude that A C E ^ f E and A C E" 1 fE, respectively, hence, by (2.203), AC ( F r T f E ) n ( E - 1 t E ) = Id.

(2.474)

By Th. 68 and (2.456), we have \/2 =

(2.475)

(R-ly\R1t\Rli-\\/,

so J\{R-loE)C(R-ly\R1t\Rii-,

(2.476) because

JIOR-1 oE) = JKR'1 o E) n ( i T Y l ^ l ^ V

(2.475)

C ( J R- 1 )*| J R fr |^|(((i?- 1 )*|^|^)" 1 |J|('R" 1 oE))

rot

l

1t

i

1

= (R- y\R \R ^\A-

(2.473)

C (R-1y\R'"\Rli

(2.474)

It follows from (2.476) and (2.472) that (X,C)£

(R-1)*^^,

as desired. To prove E\A C A, we first establish the inclusion

(2.477)

E^KEHEIJIV) C (i?-i)

34. DEDEKIND-MacNeille COMPLETION

by computing as follows. E^KEfl E|J|V) C E ^ ^ E n EKE^f-R)^) 1

C E~ |(Inii|V) 1

(2.471) (2.153)

1

= E~ |(E l~l (Id D iJ" )^)

R is domain-reflexive

C

C (R-1)*

(2.200), (2.394)

We use (2.477) to show that

J- 1 |(CR- 1 )*n(E- 1 tE)) = 0

(2.478) as follows.

= J-1\((R-1Y = J~1\((R~1)*n(R-1)*)

(2.477)

We then use (2.478) to show

J-'KR-1)" C

(2.479) as follows.

= j-1\((R-1)*n(E-1}E)

u

cj-'ifF'tE) u J~ = J"1|(ErTtE) We use (2.479) in the final calculation. E\A = EKE"1 oR^J^KR-1)*^^ l

l

1

^

(2.478) (2.473) (2.432) (2.479) (2.457) (2.162), f-assoc

C i?|(i?f i F 1 ! E ) | ( E ^ t ^ t E ) | f l ^ ^

(2.471) (2.156)

2. S E T THEORY

c (i?|i?fE)| n RV c ( i J t E ) ! ^ n R\V

Peirce's Law

C (i?tE)|(E-it-R- 1 t(EU J R|V)) n R\V

(2.458)

C (EUi?|V) n R\V

(2.153)

t-, -mon, R istransitive

C E

Th. 12 X

J

:

To show A\E~ C E~ we first notice that J ~ C i ? " 1 ^ since j c (riffljnE-'lv C E-'KVnEKE^fi?))

rot

C E^^i?

(2.153)

C V\R 1

It follows that l /~ |( J R~ 1 )*|^ fr C J - J | V C iT^VIV = fl-^V. Furthermore, we also notice that J

1V

(2..473)

)

C (RC

X

"t^KE" \R- \E) 1 1 (R~ t^lE" !-R-'fE 1

(2..471), (2.394), (2.200) Peirce's Law, |-assoc

1

c R- Ifl-'tE c R-1 tE

(2..152) R\

I, f-mon

These last two observations give us the first step in the next computation.

R-^y^nR'^v

(2.457) (2.153)

CE Use this last inclusion in the second step: AlE'1 = ( E " 1 * ^ ) ^ " 1 ! ^ " 1 ) * ! ^ ! ^ ^ " 1

C (E'^^IEI^IE"

(2.473)

1

"tfl"1)!^)^"1

(2.458)

1

(2.152)

-1

I-mon

ffl" ) \R )

(2.153) C E"

1

This completes the proof.

(2.152) D

34. DEDEKIND-MacNeille COMPLETION

117

For a more conventional proof in case R is a partial ordering, assume u is a least upper iZ-bound of X / 0. This entails that u is in the iZ-cone of every upper iZ-bound of X, so u is a lower iZ-bound of the set of upper iZ-bounds of X, that is, u £ (R('\Ril')(X). By the transitivity and range-reflexivity of R, a set is an upper .R-bound of the union of the iZ-cones of elements of X iff it is an upper .R-bound of X itself:

What we are trying to prove therefore reduces to (R-loE)(u)=R*(R1t(X)). Suppose w is in the iZ-cone of u, that is, w £ (iZ"1 o E)(it). Then (w,u) £ R. By transitivity of R, this gives us w £ iZ^(iZ^(X))). For the other direction, suppose w is R to every upper bound of X. One of those upper bounds is u, hence (w, u) £ R, i. e., w £ (iZ~1 o E) (u). Recall (p. 76) that a class R is extensional iff (iZ"1 o R) n iZ'^VliZ C Id, and E is extensional. Theorem 80. Let R CM2. (i) // iZ is transitive and extensional, then R is antisymmetric. (ii) // R is reflexive and antisymmetric, then R is extensional. (iii) If R is a partial ordering, then R is extensional. PROOF.

Proof of (i): First note that

iZ"1 n R = iz^lid n id\R c iz-^vn v|iz = .R-^viiz by |-mon and (2.115). Hence R is antisymmetric because iZ"1 niZC (iZ" 1 ffl)n(!R r T t^)riiZ" 1 |V|iZ 1

1

= (iZ" oiZ)niZ" |V|iZ C Id

R is transitive, Th. 20 (2.200) R is extensional

Proof of (ii): Since R is reflexive, we have R = iZK-R"1 n Id) and hence also R-1 = (iZnld)|iZ- 1 , so R~X\V\R = (iznid^iz-'lviiZKiZ" 1 n id) c ( ^ n id)|V|(JR-1 nid). Then, using various parts of Th. 16, Th. 17, and Th. 18, we obtain (iZ- 1 oiZ)niZ" 1 |V|iZ

c (R-1 oiZ) n (id n fl)|v|(id n R'1) = (id niz)|(iz"1oiz)|(id niZ"1) c R\(1FI jR)\\d nidKiz^tiZ)!^" 1 cizniZ" 1 C Id so R is extensional.

R is antisymmetric,

118

2. SET THEORY

Proof of (iii): A partial ordering is reflexive and antisymmetric, and so is also extensional by part (ii). 34.4. Completion of a relation. The next theorem generalizes Dedekind's method of cuts. For partial orderings it is due to MacNeille [136]; see McKenzie-McNulty-Taylor [167, p. 51,53]. We prove it for a binary relation R that is transitive, domain-reflexive, and extensional. By Th. 80(i), such a relation is antisymmetric. If R is also range-reflexive then, since R is already domainreflexive, R is reflexive and hence, because it is antisymmetric and transitive, R is a partial ordering. For an example of a transitive, domain-reflexive, extensional relation R that is not a partial ordering, choose distinct a, 6, c e V and let R={{a,a),{b,b),(a,c),{b,c)}.

Theorem 81 (Dedekind-MacNeille completion). Assume R is a set and a transitive, domain-reflexive, extensional binary relation. Let F=(R-1oE)DR-1\\/.

(2.480) Then

(i) F : Fd(R) -> Sb (Fd(R)) ~{0}, F is infective, and x G F(x) for every xeFd(R). (ii) Every element in the range of F is (R^\R^)-closed, in the sense that F\R1t\R11' = F, in other words, (R1t\Ris-)(F(x)) = F(x) for every x G Fd (R). (iii) F embeds R into the inclusion relation on (R^\R^)-closed subsets, i.e., F'^RIF C E ^ f E; if{x,y) <= R then F{x) CF(y). (iv) F preserves all least upper R-bounds and greatest lower R-bounds: for then all X e Sb (Fd(R)), ifJ2R x exists

and ifY[ X exists, then R

X) = OR"1 o E

Note that F is the domain-restriction of R^1 o E to the range of R. But R is domain-reflexive so the domain of R is a subset of the range of R. Hence F is actually just the domain-restriction of R-1 o E to the field of R. The range of R'1 o E is a subset of Sb (Fd (R)) ~{0}, so we have PROOF.

F C Fd (R) x (5*6 (Fd (Rj) ~{0}). If a: € Fd(R), then x 6 F(x) since R is domain-reflexive. F is injective since FlF'1 C ( J R- 1 oE)|(E- 1 o J R)n J R- 1 |V|(V| J R) 1

1

(2.480), |-mon

= R~ oRnR~ \\/\R

(2.425)

C Id

R is extensional

Next, * i ^ C RKR-1 o E)\R*\R^

(2.480)

34. DEDEKIND-MacNeille COMPLETION

v = R\(\do E)|iZ |i?

(2.468)

JJ

E)|iJ '|(EE-i fE

= R\(R-'

oE)|(E-

(2.463)

c R\(R-i t E ) | ( E - ME) 1 c EKE- tE) c

(2.468)

LJJ

U?

C R\(\do

hence F\I l*\R^ C

(2.481)

E.

Furthermore,

(^:oE)MI* t(EUi?-- : |v))

c (ir^EOK

(2.457)

1

C EUi?" |V

(2.153)

Using this, we prove an inclusion:

Q(i

r1 ^ n r : |v n

CE|

B^IE"1

F|^|i^l\E

1J CE| (E~ \R

oE^li^li^lE"1 oE)!^)!^^" 1 - i

f E

.

JIE"1

C R- i The inclusion just derived is equivalent to (2.482)

Fl 1

Let X := R^lR ^. From (2.481) and (2.482) we get F\X CR'1 oE.

(2.483)

We have F\X C i ? " 1 ^ immediately from (2.480) which, together with (2.483),

gives us F\X C R-1 oEf) R'^V = F. Then = Fn\/\x 1 cFn(vnF|X) X' = FnF|X|X"1

(2.456) rot

: l CFIXKX^1 nx - \F~ | F ) CFXftF-^F)

rot

CFlXlId

F is functional

= F\X so we have F ^ ^ = F\X = F, as desired.

F\X C F

2. SET THEORY

Since R is antisymmetric, greatest lower iZ-bounds and least upper iZ-bounds are unique. Furthermore, F preserves least upper bounds by Th. 79 and greatest lower bounds by Th. 78.

CHAPTER 3

General algebra In this chapter we develop enough of the general theory of algebras for subsequent purposes. Our primary reference for the general theory of algebras is Henkin-Monk-Tarski [93, Ch. 0]. 1. Algebraic structures Recall from p. 66 that, for any classes / , A, and B, we write / : A —> B or A -4 B to signify that / is a function, A is the domain of / , and the range of / is a subset of B, in which case we say that / is a function from A to B. Is possible that / , A, and B are proper classes even if / : A —> B. We also say that / is a function on A if / : A —¥ V. Using the direct product operation we can now characterize functions from A to B as follows. (3.1)

f:A-*B<*fCAxB,A2C

/|V, f~x\f C Id.

In case / : A —> A, we say that / is a unary operation on A. If / : A2 —> A then / is called a binary operation on A. If R C A x B and A, B € V, then the binary relation R determines some natural functions among the sets A, B, and their powersets Sb (A) and Sb (B). For example, if p = R* n (Sb (A) x Sb (B)), t=(RoE)n(AxSb(B)), n = (Rthen p : Sb (A) -> Sb (B) 7T : B -» Sb (A) Functions like these are used in Th. 84 and Th. 85. If a g A then t(a) is the U-image of {a} in B, that is, the subset of B consisting of elements to which a is related by R. If R happens to be a congruence relation (see p. 124) then i and it map an element to its congruence class. The various types of algebras defined below are all sets with some combination of constants, unary operations, and binary operations. The correlation between names and types is not entirely arbitrary since, for example, we present relation

122

3. GENERAL ALGEBRA

algebras as algebras of relational type and groups as algebras of group type, although groups could with some changes in the axioms be presented as algebras of Boolean type or even as groupoids. The types are displayed in Table 1. - Si is a unary algebra (or algebra of unary type) iff there are nonempty sets A and v such that 21 = {A, v) and v : A —> A. - 21 is a groupoid iff there are nonempty sets A and j3 such that 21 = {A,/3) and/3: A 2 -> A. - 21 is an algebra of Boolean type iff there exist nonempty sets A, fi, and v, such that 21 = (A, f$, v), f$ : A2 -» A, and v : A -> A. - St is an algebra of monoid type iff there are nonempty sets A, /3, and t, such that SI = {A,j3, t), f3 : A2 —> A, and t G A. - 21 is an algebra of group type iff there are nonempty sets A, fi, v, and 4, such that 21 = {A, f3, v, i), f3 : A2 —> A, v : A —> A, and i g A. - 21 is an algebra of relational type iff there exist nonempty sets A, fi, v, fi', v', and*, such that 21= (A,{3,v,8',v',i), 0 : A2 -* A, v : A -* A, $ :A2 -^ A,v' :A^ A, and i € A. In each case the nonempty set A is called the universe of the algebra St. Relation algebras are algebras of relational type, so if 21 is an algebra of relational type, we typically use Schroder's notation for the operations and let 21 = (A,+,~, ;, ",1 ! ). In connection with any given algebra of relational type, some additional elements, namely, 0!, 1, and 0, and some additional operations, namely, -, d , f, and r , are defined as follows. (3.2) (3.3) (3.4) (3.5)

0! := ¥, l:=l'+r, z-y:=z

+ y,

0 :=!'+!', zfy:=z;y,

x6:=V-x;x,

x':=V-x;x.

The element 1' is the identity element of 21, 0' is the diversity element of 21, 1 is the (Boolean) unit element of 21, and 0 is the (Boolean) zero element of 21. The operation + is called Boolean addition, the defined operation is called Boolean multiplication, the operation ; is called relative multiplication and the defined operation f is called relative addition. For every x £ A, the element xA is the domain of x, and x' is the range of x. The algebra (3.6)

» [ ( « ) : = (A,+,">,

obtained from 21 simply by deleting the remaining operations and the distinguished element, is called the Boolean part, or Boolean reduct, of 21. The relative part (or Peircean part) of 21 is the reduct (A,;, ", 1'), an algebra of group type. It could therefore be called the "group reduct", but it will in general not be a group. Any ambiguity regarding which algebra such notation refers to can be eliminated by the use of superscripts and subscripts. For example, algebras 21 and © have distinct identity elements iff l'at ^ l ' » . Two algebras are said to be similar if they are of the same type, that is, either they are both unary algebras, or both groupoids, or both algebras of monoid

2. SUBALGEBRAS

number of binary operations: 0 1 H

1—1

name of number of number of type of constants: unary operations: algebra: unary 0 0 0 groupoid 1 group 1 0 monoid 1 Boolean 0 2 relational 1 TABLE 1. Table of similarity types

1 2 of algebras

type, or both algebras of group type, or both algebras of Boolean type, or both algebras of relational type. The notions of subalgebra, congruence relation, homomorphism, direct product, and so on will be denned only for algebras of group type. An algebra of group type contains one representative for each type of operation: one binary operation, one unary operation, and one distinguished element. Appropriate modifications suffice to extend these notions to other types of algebras. It should be noted, however, that these concepts, and symbols used to denote them, do depend on the similarity type of the algebras under discussion. The appropriate meaning must therefore be determined from context. 2. Subalgebras Suppose that 21 and 53 are algebras of group type, and that, in particular, (3.7)

%={A,P,V,L),

j3:A2^>-A,

(3.8)

= {B,p/,v',i),

/3':B2^B,

v.A->A,

i e A,

v : B -> B, i G B.

The algebra a is a subalgebra of 55 iff A C B, j3 C ft, v C v', and t, = i . The same symbol denotes both the subalgebra relation and the subclass relation. Thus, for example, 21 is a subalgebra of 53 iff 21 C 53. Furthermore, if 21 C 53 then, among other things, A C B, but the converse may fail. For any subset X C A, let S
Sgm(X)

:= f]{C :XCCCA,

/3*(C2) U v*{C) C C , t £ C}.

It follows immediately from (3.9) that Sg^(X) is closed under j3 and v, so we let Gg*-21' (X) be the subalgebra of 21 whose universe is Sg^(X), and refer to 6fl ( a ) (X) as the subalgebra generated by X. In fact, if 5 = Sgm{X) then

124

3. GENERAL ALGEBRA

The universe Sg^ (X) of the subalgebra generated by X can also be described as follows. Define a sequence of subsets of A by setting Xo = X U {4} and Xn+i =XnU

P*{Xn x Xn) U v*(Xn)

whenever n e w . Then \Jneu X» = A whenever X generates 91. According to the next theorem, each element in a generated subalgebra depends on only finitely many generators. Theorem 82. Suppose SI = {A,ft,v, t) is an algebra of group type and X C A. Then, for every element a E Sg^(X), there is a finite set V C X such that a e Sgl*>{V). PROOF. Let F be the subset of A consisting of those elements of SI for which the conclusion is true, that is, F = {a:a€A,3v(VCX,\V\
Sgm(V))}.

We claim that F is a subalgebra of 21 that contains X. First, 1 € F since 0 C X, |0| = 0 < u» and 1 g Sgm{$). Next, if x g X then xeF since {a;} C X, \{x}\ = 1 < w, and x € Sg^({x}). Finally, if x,y g F then there are finite subsets V,W C X such that x g Sgm(V) and » € Sfl (a) (W), hence v(x) 6 55 ( a ) (F), j8(r,y) £ Sgm(V U W), where V U VF is also finite. Thus F is a subuniverse of 21 that contains X. It follows that Sgm(X) CF.

3. Congruence relations and quotients We say that R is a congruence relation on an algebra of group type SI (satisfying (3.7)) iff R C A2 C iJ|V, iZ is an equivalence relation, v " 1 ^ ^ C .R, and (9~1[(P|J2[P"1 nQ|J2|Q—1)|/3 C R. Let CoSl be the set of congruence relations on St. Consider an arbitrary congruence relation R € CoSt. Recall from (2.396) that A/R is the set of equivalence classes of elements of A, i.e., (3.11)

A/R := {(]J«E)(a) : a G A} = ( # 0 E)*(A),

Let (3.12)

i/fl:=(i?oE)(t),

(3.13)

« / # := E"~V|(i*o E) n (^/-R)2,

(3.14)

P/R := ((PIE-^P- 1 ) n (QIE-^Q-^JI^Kiio E) n {A/R)\

(3.15)

a/fl := (A/R, P/R, v/R, i/R) .

In this context there is clearly a considerable notational overload placed on the expression "/.R", whose meaning must be understood according to the nature of the object denoted by the symbol placed in front of it. Such overloading is, however, convenient and customary, especially for (3.11) and (3.15). This custom has been extended here in (3.12)-(3.14). Although v/R and fi/R have straightforward definitions as relations according to (3.13) and (3.14), what is required for the proof of the next theorem is to show that v/R and {3/R are actually functions.

4, HOMOMORPHISMS

138

Theorem 83. If 21 = (A,(3,v,i) is an algebra of group type and R 6 Co2l, then t,/R 6 A/R, v/R : A/R -* A/R, P/R : (A/Rf

-> A/R,

Si/R is an algebra of group type. For any congruence relation R £ CoSL, the algebra 21/.R is called the quotient algebra of 21 modulo (or, with respect to) R. 4. Hotnotnorphisms For two algebras of group type SI and 93 (satisfying (3.7) and (3.8)), we say that ft is a hotnotnorphism from 21 to 2$ if h : A —> B (see p. 66 and p. 121), h{i) = t', v\h = h\v', and ^ / ~ 1 |(P|ft- 1 [P~ 1 n Q I A - ^ Q " 1 ) ! ^ C Id.

(3.16)

Condition (3.16), expressed differently, and perhaps more familiarly, is that V o V 6 (a,6e A =

h(p{a,b))=tf(h(a),h{b)))-

We say that ft is a hotnotnorphism from St onto OS if ft is a homomorphism from 21 to !8 and the range of h is B. For any two algebras 21 and 03, let Hom{SL, 9$) be the set of homomorphisms from SI into 93, and let iJo(Sl, 93) be the set of homomorphisms from St onto «S. Note that if 0 ^ Ham{% 33) or 0 £ Ho(% 95) then SI and 93 must be similar algebras, although the converse need not hold. According to the next theorem, every quotient algebra is a homomorphic image of St. Theorem 84. If R is a congruence relation on St and h = (Ro E) f~l (A x V), then (i) h-.A^A/R, (ii) h is a homomorphism from SI onto 21/iZ, and (iii) felft"1 = ii. PROOF. We only prove part (iii). First we show that fcl/T1

(3.17)

^(RoR-^nA2

as follows: hlh'1 = {{Ro E) n (A x V))|((i?o E) D(Ax V))" 1 = ((.Ro E) n (A x V ) ) ! ^ - 1 oiJ- 1 ) n (V x A)) = (.Ro EJKE"1 oR'1)

n (A x V) n (V x A)

by (2.362), (2.364), (2.113), (2.114) = (RoR'1) n A2

by (2.425), (2.373)

3. GENERAL ALGEBRA

From our assumption that R £ Co2l, we know that R is transitive and symmetric, and (3.18)

A2 C R\V,

(3.19)

RCA2.

Since R is transitive and symmetric, we conclude by (2.219) that R\(RoE) C RoE, so 1 1 1 (R o E ) - ! ^ C (ROE)' . (3.20) With the help of this last equation we can show that RCRoR-1 (3.21) as follows. R = R-1n(RoE)\V symmetry of R, (2.420) 1

1

C (iioE)|(Vn(ii<>E)~ \R- ) C (i?oE)|(i?oE)" 1

rot.

1

(3.20) 1

= (RoE)\(E- oR- ) = RoR~1

(2.425)

It follows from (3.17), (3.19), and (3.21) that R C ft|/i-1. For the opposite inclusion, !we have h\h~l C (RoR-1)!! R\V (3.17), (3.18) C i2|(i?~1|(i2« R'1)) 1

C RIR-

rot (2.214) R is an equivalence relation

CR

We say that h is a homomorphism on 21 iff h is a function on A and /i|/i x is a congruence relation on 21. Let Ho2l be the set of homomorphisms on 21. We have defined two kinds of homomorphisms, a homomorphism on an algebra and a homomorphism from an algebra to (or onio) another algebra. The two notions of homomorphism are essentially the same. Any homomorphism from an algebra 21 to a similar algebra 55 is a homomorphism on 21, that is, flo(2l,<8) CfloSl. Conversely, if ft is a homomorphism on an algebra 21 = {A, j3, v, i) satisfying (3.7), then we will obtain an algebra 03, such that h is a homomorphism from 21 onto 53, if we let (3.22)

B:=h*(A),

(3.23)

0 := (Pl/r'ip- 1 nQl/r'lQ

(3.24)

v

(3.25)

i := h(i),

:=hTx\v\h,

5. FILTERS AND IDEALS (3.26)

03

127

:=(B,P',V',L').

We say that h is an isomorphism from 21 onto 03 if h is a homomorphism from 21 to 03, 7i is injective, and the range of h is B (h maps 21 "onto" 03), in which case we say 21 is isomorphic to 03 viaft,and symbolize this fact by writing 21 = 03, h

or, taking note of the isomorphism that makes this true, by writing 21 = 03. An embedding is a homomorphism that is injective but not necessarily onto. There is an embedding of 21 into 03 iff 21 is isomorphic to a subalgebra of 03 iff 21 =|C 03. The algebra 21 is said to be simple iff 21 has exactly two congruence relations, namely A1 and A2. If 21 is simple then A contains at least two elements, for if A is a singleton, then A1 = A2 and 21 has only one congruence relation. A homomorphism is said to be nontrivial iff it is neither a constant function nor a bijection, for then we may characterize simple algebras as nontrivial algebras that have no nontrivial homomorphisms. A nontrivial algebra has more than one element, and every such algebra always has at least two homomorphisms: one that collapses the algebra onto a 1-element algebra, and another, namely, the identity relation, that maps each element to itself. The next theorem (sometimes called the "first homomorphism theorem") shows that every homomorphic image is isomorphic to a quotient algebra. Theorem 85. Suppose f 6 i?o(2l, 03) is a homomorphism from 21 onto 03. LetR=f\f-1, h=(R*E)n(AxV), and g = E'^f n ((A/R) x V). Then (i) R £ Co2l is a congruence relation on 21, (ii) h E Ho (21,21/R) is a homomorphism from 21 onto 21/R, (iii) g is an isomorphism from 21/R onto 03, (iv) f = h\g. 5. Filters and ideals We say that T is a filter on a set U € V iff T is a subset of 56 (U), contains U, is closed under intersection, and contains every subset of U that contains one of its elements. Again, more formally, T is a filter on U if the following statements hold for all classes Y and Z: (i) T
128

3. GENERAL ALGEBRA

of complements of T is an ideal on U. We say that J- is an ultrafilter on U if, for every Y GU, either F € J o r [ f ~ F 6 f . A filter J 7 is a maximal filter if there is no proper filter Q on U such that T C Q. A fairly straightforward exercise is to show that a filter is a maximal filter iff it is an ultrafilter. Similarly, I is a maximal ideal on U if there is no proper ideal J on U such that X C J An ideal is maximal iff, for every Y C U, either Y 6 J o r [ / ~ 7 € J . For every x £ [/, the set {F : x £ F C U} is an ultrafilter on U, called the principal ultrafilter generated by x. An ultrafilter that is not a principal filter is called a nonprincipal ultrafilter. The Axiom of Choice (in the form of Zorn's Lemma or the Well-Ordering Th. 72) can be used to prove the Maximal Ideal/Ultrafilter Th. 86. Theorem 86 (Maximal Ideal/Ultrafilter Theorem). Every proper filter on a set U £ V is contained in an ultrafilter on U. Every proper ideal on U £ V in contained in a maximal ideal on U. 6. Products of algebras In this section we define the direct product of two algebras, the direct product of an indexed system of algebras, and some related notions. Assume 21 = (A,f3, v,i) and 58 = (B,/3',v',t') are algebras of group type. Officially, 21 and 03 are actually ordered pairs. For example,

and the constituent parts of 21 can be obtained by applying projection functions, in the sense that

21 = (((A,0) ,v),i) = <«P|P|P(2l), P|P|Q(2l)>,P|Q(2l)>, Since the ordered pairs 21 and 55 are two-element sets, 21 x 55 is a particular fourelement set, according to the way "x" has been defined so far. However, this symbol is also commonly used for the direct product of algebras, and this practice will be followed here. The direct product of 21 and 23 is (3.27)

21 x 03 := {A xB,/3",

v",i"),

where 1

n Q|«/|GT\ i

^

i

n

(P|Q|P-!

n

Among the consequences of these definitions are alternative ways of presenting v" and /3", such as v"((a,b)) = (v(a),v'(b))

whenever a £ A and b £ B,

v" = {((a, b), (c, d» : (a, c) £ v A (b, d) £ v'}, v" = ((v(a),v'(b)):ae P"({a,b), (x, y)) = (j3(a,x),P'(b,y))

A, b e B) , whenever a,x £ A and b,y £ A,

6. PRODUCTS OP ALGEBRAS

f

= {{{a, 6}, {x,y} , £ ,3'},

f

= «/?(a, a),/?(&,»)) : a, 6 £ A Ax,» £ fl) .

129

There are homomorphisms that project 21 x 95 onto its factors SI and 2$. Indeed, P l~l (A x .B)|V is a homomorphism from 21 x 9} onto 21, and Q n (A x B)|V is a homomorphism from 21 x 2$ onto 25. We say that SI is a system of algebras of group type indexed by I if 0 / J £ V, there is a set J K, and every element of K is an algebra of group type. Suppose Si is a system of algebras of group type indexed by J. If A = 2l|P|P|P, P = 2l|P|P|Q, v = 2l|P|Q, and i = 2l|Q, then A, /3, v, and 1 are functions on / , that is, A : / —> V, j3 : I — V, v : I —> V, and i : I — V, so if we adopt subscript notation, and write Ai, and tj £ Ai. The d i r e c t p r o d u c t of 21 is

(3.28)

Hm-.

where (3.29)

JjA::={ff : ACo-|E,
(3.30)

w" := (<«i(ffi) : i £ I) : a £

(3.31)

^ " := (<#(<*, rO : £ J) :
For a more conventional presentation of the direct product, it is perhaps worth noting that a e ]]A iff a : I —> \JieI Ai and ai £ Ai for all i £ / . The two equations defining v" and j3" assert that, for every

For each i £ / , "evaluation at i" projects JJ21 onto SLi, that is, {at : CJ £ a homomorphism from n S t onto 2lj (its range is Ai). An algebra 9} of group type is said to be a subdirect product of a system 21 of algebras indexed by I if 93 C FJ 21 and {ai : a £ F[ A) is a homomorphism from F| SI onto SU for every t £ J, and not just into, as would be the case if the algebra 95 were a subalgebra of the direct product of the system SI. A longer but more illuminating notation for the direct product is

The algebra 93 is said to be semisimple if 05 is isomorphic to a subdirect product of simple algebras. Now just assume 21 is an algebra of group type (rather than a system). The algebra 21 is subdirectly indecomposable iff St is nontrivial (has at least two elements) and A1 C f]{-^ A1 C R £ CoSl}, that is, there is a uniquely determined smallest congruence relation on SI that is not the identity relation on A (see Henkin-Monk-Tarski [93, 0.3.50(ii), 0.3.52]). The algebra SI is directly

3. GENERAL ALGEBRA

indecomposable iff 21 is nontrivial and for all R,S(z Co2l, if R fl S = A1 and R\S = A2, then either R = A1 and 5 = A2, or else R = A2 or 5 = A1 Henkin-Monk-Tarski [93, 0.3.25]. Every subdirectly indecomposable algebra is directly indecomposable, but the converse is false. For more on these notions and, in particular, theorems that explain the choice of terminology, consult HenkinMonk-Tarski [93, Ch. 0]. For example, an algebra is subdirectly indecomposable iff it is not isomorphic to a nontrivial subdirect product. Consider again a system 21 = (2li : i E I) where every 2ti is an algebra of group type, so that if A = 2t|P|P|P then 2li = {At, ) for all i e I. For any filter T on I, let (3.32)

=£:= {{a, T) : a,r e~[[A, {i : at = n, i E 1} E T}.

Then = J is a congruence relation on rj2l, a n d the quotient algebra rj2l/=a, also denoted simply YlVL/F, is called a reduced product. Elements of the reduced product are equivalence classes a / = J with a E A, also sometimes denoted simply ajT. If T is an ultrafilter, then fj 21/J7 is called an ultraproduct. If T is a nonprincipal ultrafilter, then FJ 21/J7 is called a nonprincipal ultraproduct. 7. Operators S, H, I, P, Up Let if be a class of similar algebras. We will define SK, HK, \K, PK, and Up-ftT so that they are the classes of algebras that are, respectively, subalgebras of algebras in K, homomorphic images of algebras in K, isomorphic to algebras in K, isomorphic to direct products of systems of algebras in K, and isomorphic to ultraproducts of systems of algebras in K. The definitions of SK, HK, \K, PK, and UpK given below follow Henkin-Monk-Tarski [93, 0.1.8, 0.2.6, 0.3.1(iii), 0.3.62(ii)]. (Note the use of "e" in place of "E", and that "C" denotes the subalgebra relation, not the subset relation.) (3.33)

SK := {21 : 21 C|e K},

(3.34)

HK := {<8 : Ho(% <8) ^ 0},

(3.35)

\K := {21 : 21 ^|<E K},

(3.36)

PK := \{[[
(3.37)

UpK := \{Y\ ^IT

0

K},

I e V, 03 : / -> K, T is an ultrafilter on / } .

The inclusion of I in the definition of P insures that if \K = K then HK = K, SK = K, and PK = K. For further remarks on this issue, see Henkin-MonkTarski [93, pp. 83-4]. Note that if HK = K or PK = K, then \K = K. However, it is possible that SK = K and \K ^ K. The next theorem expresses some of the basic properties of the operators S, H, I, P, Up. Results summarized in the next theorem can be found in Henkin-Monk-Tarski [93, 0.1.14, 0.2.13, 0.2.19, 0.3.12, 0.3.69]. Theorem 87. Assume K and L are classes of similar (3.38)

K C SK = SSK,

8. ASSEMBLY LEMMA

(3.39)

A" C L => A" C SL <^> SAT C SL,

(3.40)

ATC IAT = IIATC HAT = HHAT= HIAT = IHAT,

(3.41)

K C L => K C HL <^> HAT C HL,

(3.42)

AT C L => K C IL <£> IK C IL,

(3.43)

SHAT C HSAT,

(3.44)

SIAT = ISA",

(3.45)

A" C IA" C PAT = PPK = P\K = \PK,

(3.46)

PSK C SPA",

(3.47)

PHATCHPAT,

(3.48)

AT C L => AT C PL «* PA" C PL,

(3.49)

AT C IA" C UpA" = UpUpA" = UplAT = lUpAT C HPA",

(3.50)

UpSAT C SUpAT,

(3.51)

UpHAT C HUpAT,

(3.52)

K C L => K C UpL UpAT C UpL.

131

For complete details regarding these operators on classes, see Henkin-MonkTarski [93, §0.1-0.3, 0.3.16], McKenzie-McNulty-Taylor [167, Fig. 4.9], and Pigozzi [201]. Let K = {Q} where Q is the algebra whose universe is the set of rational numbers and whose operations are the ordinary addition and multiplication. It is easy to show that every algebra in HAT is either a 1-element algebra or is isomorphic to Q, and that every subalgebra of Q is either a 1-element algebra or is infinite. Hence SHAT contains no nontrivial finite algebras. On the hand, there are many nontrivial finite algebras in HSAT, so SHAT C HSAT. Suppose K is the class of all groupoids (i.e., algebras of groupoid type, having but one binary operation) that have either one or two elements. Then SAT = HAT = K C PAT. Every algebra in PAT is either finite or uncountable. On the other hand, SPAT does contain some countable algebras. Hence PSAT = PAT c SPAT. There are also examples for which PHAT C HPAT.

8. A s s e m b l y L e m m a The Assembly Lemma embodies a frequently used observation. Suppose K is class of similar algebras closed under subalgebras and direct products, that is, SAT = PAT = K. By Th. 87, an equivalent way to express this condition is that SPAT = K. According to the Assembly Lemma, to embed a given algebra 21 into an algebra in K it suffices to show that every pair of distinct elements of 21 are mapped to distinct elements by some homomorphism from 21 to an algebra in K. T h e o r e m 88 (Assembly Lemma). Let 21 be an algebra with at least two elements. Assume, for each pair of distinct elements x,y G A, that there is an algebra 93a,j, and a homomorphism hx,y from 21 into *Bx,y such that hx,y(x) / hx,y{y)-

132

3. GENERAL ALGEBRA

Then 21 is isomorphic to a subalgebra of the direct product

n

In fact, if h is defined for every z € A by (3.53)

h(z) := {hx,y(z)

: x,y G A, x /

y),

then h embeds 21 into the direct product.

9. Clones Use P and Q to define additional projection functions for 2 < n € u, by setting

(3.54)

P o := Q, Pi := P|Q, P 2 := P|P|Q, • • • , Pn := P|P|P| . .. |P| Q, • • • n

There is a notational clash here, for the symbol P is used to denote both a projection function and a function P : w Sb (V) whose range contains projection functions. Given a set B and a set F C Sb (B2) of binary relations on B, define the unary clone generated by F to be closure of F under relative multiplication, namely, P|{G :FCGCSb

(B2) , VCTVT(CT, T € G = a\r € G)}.

If every element of F is a unary operation on B, that is, if VCT ( a : B ^ B), then every element of the unary clone generated by F is a unary operation on B. This is easy to prove by induction, and uses the fact that the relative product of functional relations is functional. Now suppose we have a triple consisting, in order, of a set of constants, a set of binary relations, and a set of ternary relations, all on B, that is, Fo C B, Fi C 56 (B2), and F 2 C 56 (B2 X B) = Sb (B3). For every positive integer n, 0 < n € w, the n-ary clone generated by (Fo, Fi, F 2 ) is the smallest subset G of 56 (Bn x B) = Sb (B" + 1 ) that contains the n-ary projection functions, contains the n-ary constant functions taking a value in Fo, and is closed under composition, that is, G is the smallest set with these properties: {PiDBn+1 :i
10, FREE ALGEBRAS

133

PROOF. The key observation in an inductive proof of this fact is that if er, T, fi are functional, then (
(HP-1 n r l Q ^ ' W I ) 1

1

1

1

( K I J I ) ) ^

1

|-assoc

C ^~ |(P|P~ n QIQ" )!^

or and r are functional (2.313)

C Id

i9 is functional. D

The clone generated by {i
{i,v(x),?3(y,z)

(3.56)

|{*,w(z)>0(v,*)}l = 3,

(3.57)

v(x) = v(y) => x = y,

(3.58)

/S(M>, X) = / % , z) =s- w = y A x = z.

Then 21 is absolutely freely generated by X.

3. GENERAL ALGEBRA

PROOF. Let 03 = {B, (]', v', t'} be any algebra of group type and let / : X — B. Then / C X x B, so / is a subset of the universe of the direct product 21 x 03 and generates a subalgebra h in that product, i.e., h = 6cj<-alx<8' (/). Then eS(
(3.59)

Ra(h) =

(3.60)

Do (h) = 6 0 ( 2 l ) (Do (/)) = 6g (21) (X) = A.

Next we will use the hypothesized conditions on i, v, and /3 to show that h is functional. For this it suffices to prove by induction that every pair (a, 6) in h has this property: (3.61)

(c,d) eh,

a = c => b = d

Suppose first, as the base case in the induction, that (a, 6) is a generator of h, that is, (3.62)

{a,b)ef.

This implies a £ X, a fact we will use several times. For a proof of (3.61) in case (3.62) holds, assume (c, d) £ h and a = c. We wish to prove b = d. Now (c, d) is either a generator of h or is in the image of h under the operations of 21 x OS. If (c, d) is a generator then it is in / , in which case we have (a, 6) £ / , (c, d) £ / , a = c, but these conditions imply b = d simply because / is functional. So we may suppose that (c, d) is in the image of h under the operations of 21 x 03. This produces three (impossible, as we will see) cases: (c, d) may either be the pair (i, (,'), or it is in the image of h under the unary operator of 21 x 03, or it is in the image of h under the binary operator of 21 x 03. Now (c, d) cannot be the pair (t, t'), for this would entail t = c = a £ X, contrary to (3.55). It also cannot happen that (c,d) = (v(x),v'(y)} for some (x,y) £ h, for then v(x) = c = a 6 X, but this is also ruled out by (3.55). Similarly, (c,d) cannot be in the image of h under the binary operation of 21 x 03. This completes the proof of (3.61) in case (3.62). Now suppose, as first of two inductive cases in our proof by induction, that (a,b) has the property (3.61). We wish to show that (v(a),v'(b)) is also a pair that satisfies condition (3.61), i.e., that (c,d) £h, v(a) = c => v'(b) = d. Accordingly, let us assume that (c, d) E h and v(a) = c and try to prove v'(b) = d. Again (c, d) is either a generator of h or is in the image of h under the operations of 21 x 03. If (c, d) is a generator then c £ X, but then v(a) = c £ X, contrary to (3.55). Similarly, (c, d) cannot be (t, t'}, for this would contradict (3.56) by requiring i = v(a). Similarly, it follows from (3.56) that (c, d) cannot be in the range of the binary operation of 21 x 03. Suppose (c, d) is in the image of h under the unary operation of 21 x 03, and, specifically, that (c, d) = {v(x),v'(y)} for some (x, y) E h. Then v(a) = c = v(x), so a = x by (3.57). We may now apply (3.61) to the pair (x, y) £ h and conclude that b = y. But we already have d = v'(y), and this last equation gives us v'(b) = v'(y), so v'(b) = d, as desired.

10, FREE ALGEBRAS

138

If (e, d) is in the image of h under the binary operation of SI x 58, say (c, d) = (/3(«,a;), 0'(w,y)} for some (v,w), (x,y) € ft, then v(d) = c = 0(v,x), contrary to (3.56), so, in fact, this case cannot occur. For the second of our two inductive cases, assume that the pairs {a, 6} and (a',b') have the property (3.61). We wish to show that (0(a,a'),£'(&,&')) also satisfies (3.61), i.e., (c,d) £ h, ji{a,a') = c => /S'(6,&') = d. Assume that (c, d) € ft and /3(a, a') = c. We wish to prove ff (b, b') = d. Now {c, d} cannot be a generator, since otherwise fl(a,a') = e E X, contrary to (3.55), and (c, d) cannot be {i, i'), for this would contradict (3.56) by requiring i = (3(a, a'). It also follows from (3.56) that {c, d) cannot be in the range of the unary operation of 21 x 9$. Therefore (c, d) is in the image of h under the binary operation of 21 x !8, say (c,d) = ((3(v,x),l3'(w,y)) for some (v,w), {x,y} € ft. Then f}{a,a') = c = /?(«,»), so a = v and a' = a; by (3.58). By applying (3.61) to the pairs («, w), (a;,!/) £ ft we get b = it? and &' = y. These two equations imply /3'(&, b') = {3'(w,y) but we already have d = fi'{w,y), so d = fi'(b,b'), as desired. This completes the proof of (3.61) for both inductive cases. Since h is functional, contains / , is a subalgebra of SI x 93, and has domain A by (3.59), it follows that ft is a homomorphism from 21 into 03. There are several ways to construct absolutely freely generated algebras; see Henkin-Monk-Tarski [93, pp. 130-1]. We will use the following particular method adapted from Henkin-Monk-Tarski [93, 0.4.19(i)]. For every ordinal a, let (3.63) Fra := f]{U : a C U,%Vy(m,y G U => {0,0}, (1, (x)} , (2, (m,y)) £ V)}, and define the algebra $ta of group type by (3.64)

&a:={Fra,0,v,t)

where* = (0,0), v(x) = (1, (a;)), andp(x,y) = (2, (a;,j/}} for alia;,?/ g Fra. Notice that i and every element in the range of either v and j3 is an ordered pair, but no ordinal is an ordered pair, so Ra09) n a = Ra(t))na

= { ( } n a = 0.

The unicity property (2.95) of ordered pairs implies that # t a satisfies the sufficient conditions in Th. 91 for an algebra to be absolutely free. We therefore have the following theorem. Theorem 92. ^ r a is absolutely freely generated by a. The next theorem asserts the existence of algebras of group type that axe absolutely freely generated by an arbitrary set. Similar theorems hold for all other types of algebras. Theorem 93. For eneri/ nonempty set X £ V there exists an algebra of group type that is absolutely freely generated by X.

136

3. GENERAL ALGEBRA

PROOF. Choose an ordinal a for which there is a bijection A : a —> X. Let Y = {X} x (Fra ~ a) and A = X UY. Note that X and Y are disjoint, a consequence of the Axiom of Regularity, for if x £ X fl Y then there is some w £ Fra~a such that x = (X,w) £ X, but then X€\£\£X. This contradicts a consequence of the Axiom of Regularity, that there can be no 6-loops. Next, imitate the structure of $ta on A by treating elements of X as if they were in a, and elements of Fra ~ a as if they were in Y. Specifically, for all x, x £ X let /3(x,x') = (X, (2, (A(a;), A(a;')>» and let v(x) = (X, (1, A(a;)>). Let t := (X, (0, 0)). Then 21 = (A,/3, v,t) is the desired algebra. In view of Th. 90, we refer to $va as the absolutely freely a-generated algebra of group type. This algebra exists (is nonempty) even when a = 0, since every algebra of group type has a constant (an "operation of rank 0"). This applies as well to monoids and algebras of relational type, but does not apply to unary algebras, groupoids, or algebras of Boolean type. (Here one may choose to allow algebras with empty universes.) The elements of Fra are called terms, but if a < w and we wish to emphasize that the set of variables is restricted to the first a of them, we will refer to terms in Fra as a-variable terms. The ordinals in UJ are, in this context, called variables. Note that

Fra = | J Frp,

Pairs of terms are called equations, and pairs of a-variable terms are called avariable equations. An equation e = (to,ti) is valid in (also true in) an algebra 21 = (^4, /3, a, L) of group type if for every function a : n —> A called, in this context, an assignment, the unique homomorphism a' : Frn —> A that extends the assignment a has the property that o'(io) = a'{ti). Write 21 |= e if e is true in 21. Consequently, 21 |= e <=> h(t0) = ft(ii)for every ft 6 Hom{Fr^,iA). For every term t £ Fra, define free(t) to be the smallest subset of a that generates a subalgebra containing t, and refer to free(t) as the set of free variables of t. By Th.82, ||free(t)| < to. Theorem 94. Let n < u. Consider an algebra 55 = {B,/3',v', if) of group Let E be the type. Let Gn he the n-ary clone generated by smallest relation such that - GS, - (k, P/b n Bn+1) € E for every k < n, and - ifto,ti 6 Frn, TO,TI E Gn, and {to, TO) , (ii,ri) 6 E, then G E and (P'(to,U), (TO]?-1 n TiIQ"1)|/3'> £ E. Then (i) E is a function that maps the terms in Frn onto the elements of the n-ary clone Gn: E:Frn^ Gn,

10, FREE ALGEBRAS

137

(ii) An equation {to,ti} is valid in the algebra 33 iff the terms in the equation determine the same element of the clone Gn, that is,

»H*o,*i> iff E(t0) = Efa). (iii) For every assignment a : n —¥ B and every term to € Frn, if a' is the unique homomorphism extending a, then a (to) = E(to)(a(n - 1),... ,a(0)). If 0 ^ n < w, then for any n-ary term to € Frn in the absolutely freely ^-generated algebra #r n of group type, and any algebra 25 of group type, we will write £(f in place of E(ta), where E is the evaluation function described in Th. 94. Thus, for example, the equation (*o,ti) is valid in 25 iff i(f = *? Suppose we have a set of equations S C Fr^ and a class if of algebras of group type. We will write 9$ |= £ to mean that every equation in S is valid in 33 and K |= £ to mean that 2$ |= £ for every 2$ € if. Let Grp be the class of algebras of group type. A class if C Grp is said to be an equational class if there is an equational axiomatization for K, that is, a set £ C (.Frw)2 of u-variable equations such that K = {33 : 33 1= S, © £ Grp}. if is a finitely based equational class if there is a finite equational axiomatization for K. We say that K C Grp is a variety if K is closed under the formation of subalgebras, homomorphic images, and isomorphic copies of direct products, that is, K = SK = HK = PK. Birkhoff's HSP-Theorem is next. For more about this and related theorems, consult Henkin-Monk-Tarski [93, §0.4] or McKenzie-McNulty-Taylor [167]. Theorem 95 (Birkhoff [31, 228]). Suppose K is a class of similar algebras. The following statements are equivalent: (i) K is an equational class, (ii) K is a variety. PROOF, (i) =^ (ii): Assume if is an equational class. Then there is some set of equations E such that K (= S and, for any class L C Grp,

(3.65) L 1= S =}- L C K. It is straightforward to prove, from K \= S, that HK \= E, SK \= E, and PK 1= S. These last three statements imply, by (3.65), that HK C if, SK C if, and Pif C if. Since the opposite inclusions always hold, if is a variety. (ii) => (i): Assume if = Sif = Hif = Pif. Let E be the set of equations that are true in every algebra in if, i.e., E := {e : e 6 {Fraf, K |= e} Suppose that 21 G Grp is a algebra such that 21 |= S. By Th. 93, there is an algebra $t^ 6 Grp that is absolutely freely generated by A. Let I be the set of all congruence relations on ^r^ with the property that the correlated quotient algebra is in if. (3.66)

/ := {C : C G CodvA, g t A / C G if}.

138

3. GENERAL ALGEBRA

The direct product, over / , of all these quotient algebras is, by its construction, in PK, but PK = K, so

V : = I I ^AIC e K.

(3.67)

Gel

Next we single out the subalgebra of this direct product generated by those elements that are correlated in a natural way with the elements of A by a function f : A —¥ P. In particular, if a e A then the element of the product correlated with a is the function :CeI). /„ := (a/C Let 6 := &Qm({fa :a€A}) = S 0 ( w ( i i a ( / ) ) . Note that 6 G S{*P} C SK = K. Since $vA is absolutely freely generated by A, the function / from A to elements of %5 extends to a homomorphism h from $tA to 6 . Note that h maps $tA onto <5 since the range of / generates <5. Next we prove by induction that for every x G FrA, (3.68)

h(x) = (x/C : C G J ) .

If a G A then h(a) = fa = (a/C : C € I), so every element of A has the requisite property. Next we show elements with the desired property form a subalgebra. We only show closure under the binary operation of $xA. Proofs for the unary operation and constant are similar. Let fl^TA be the binary operation of $xA and let ^ be the binary operation of ty. Assume, as inductive hypothesis, a\ and
h(fXA{a1,a2))

= j3 (h(a\), h(a2))

ft

is a homomorphism

= / ^ ( ( a i / C : C G I), (a2/C : C G I))

inductive hyp.

= (pStA/o(ai/C,

(3.67)

a2/C) :C el)

^A (a1,a2)/C

:Cel\

def. of quotient algebras

= (x/C : C G / } . To complete the proof it suffices to show that 21 is a homomorphic image of <5, since this implies 21 G H{S} C HK = K. Since $xA is absolutely freely generated by A, the identity map from A to itself extends (uniquely) to a homomorphism g from $tA onto 21. The situation is this: ©^ -

&A

- ^ 21

We need only show that g can be factored through h, i.e., that h~1\g is a homomorphism from 6 onto 21. We begin by showing that h~1\g is functional. Suppose two elements x and y of $xA are sent to the same place by h, that is, h(x) = h(y). By (3.68), this

10, FREE ALGEBRAS

implies that x/C = y/C for every congruence relation C G I. What remains is to use the hypothesis that 21 satisfies every equation valid in K to show that g{x) = g(y). Toward this end, let V C A be a finite subset of A such that x and y belong to the subalgebra of 5^A generated by V, m,yeSgiStA)(V).

(3.69)

Let n = | y | G w and pick any one-to-one correspondence f} : V — n from V onto n. Since t) is injective and onto, rj~1 : n —¥ V is bijection from n onto V. Extend r) to a homomorphism p from 3 ^ into $ t u by choosing p to be the unique homomorphism that includes T)U((A~V)x{0}). Extend rf1 to a homomorphism from $ta into %XJL as follows. Since A ^ 0, we may choose some a 6 A, and let q be the unique homomorphism from $zu into $t A that extends the function t)~1 U ((w ~ n ) x {a}). Thus we have the following situation.

3* n C ^

4- ^r A 4 &„ -% $xA,

rjCp,

tj-1 C g.

Next we explain why q(p(x)) = E and q(p(y)) = y. Let ^ = {a : a E FTA, q(p(a)) = a}. Then F C Z for if « £ F then, by the definition of p, p(v) = r)(v) < n, so q(p(v)) = ^(piv)) = r]'1 (r](v)) = v, and if a, 6 6 Z, StA 9tA then qW (a,b))) = /3 (9(p(a)),9(p(6))) = Ps*A(a,b) so / 3 ^ ( a , 6 ) G ^ . (StA) Similarly v (a) e Z and t W t j t ) £ Z. Thus £ D Sg W t A ) (F). By (3.69), so g(p(a;)) = x and g(p(»)) = y. xyyeZ, Let e = (p(a;),p(y)}. We show next that every algebra in K satisfies e. Consider an arbitrary © G K and suppose k is a homomorphism from 3 t u into SB. Then fc maps $ t u onto some subalgebra SB' of SB. However, since SIT = K, we conclude that OS' € K. Then p\k is a homomorphism from $t A onto SB'. Let

C = p\k\((p\k))-1. ByTh.85,

95' <* St^/C. This implies, by (3.66), that C £ I. It was proved earlier that x/C = j//C for every congruence relation C £ I. Hence

{x,y)eC=p\k\k-1\p-\ which implies Thus we have shown that the equation e is in k\k-1 whenever & is a homomorphism from yrw into an algebra in K. Therefore, K \= e. It follows that e € S, hence 211= e by our hypothesis that SI |= S, But q\g is a homomorphism from 3"rw into 21, so e 6 q\g\{q\gTA = glfllff"1^"1- This implies (x,y) = (q(p(x)),q(p(y))} € gig'1, hence g(x) = g(y)- This completes the proof that h~1\g is functional. It remains to check that h~x \g is a homomorphism from & onto 21, so that St is consequently a homomorphic image of 6 € K, hence 21 € K. Notice that the construction of © depends on 21. For any class K of similar algebras and any ordinal a, let

140

3. GENERAL ALGEBRA

where

C = f]{R : R € Co$ra, $ra/R e ISK}. We refer to 3taK as the free algebra over K with a generators. For example, 5TaRRA is the free representable relation algebra with a generators. Furthermore, if 21 is any algebra similar to the algebras in K and X C A is a set of elements of 21, we say that X if-freely generates 21 if 21 = &g^ (X) (X generates 21) and, for every 03 € K and every function / : X -¥ B, there is a homomorphism h € Horn (21, 03) such that / C h. These definitions are taken from Henkin-Monk-Tarski [93, 0.4.19(ii), 0.4.23]. By Th. 103 and Th. 107 below, PRRA = RRA = SRRA. Therefore, because of the following theorem, the free RRA with a generators is a representable relation algebra. Theorem 96 (Henkin-Monk-Tarski [93, 0.4.22]). $xaK G SPK. The free RRA with a generators is constructed in Th. 553 below as an algebra of equivalence classes of formulas in a first-order language in which there are no constants or function symbols, all the relation symbols are binary, and the number of binary relation symbols is a. The algebra constructed in Th. 553 is RRA-freely generated by the equivalence classes of atomic formulas of the form voRvi. The next theorem is Los's Lemma for equational classes. Much stronger versions can be found in the literature. For example, "finitely based equational class" can be replaced by "axiomatizable by a finite set of first-order sentences" or by "axiomatizable by a finite set of Ei-sentences". They all have essentially the same proof. Theorem 97. Suppose K is an equational class of similar algebras. If there is an algebra in K that is an ultraproduct of algebras not in K, then K is not a finitely based equational class. We derive a contradiction from these assumptions: S is an equational basis for K, S is finite, 21 = (21; : i € w), 21, £ K for every i € to, T is an ultrafilter on w,

PROOF.

(i) (ii) (iii) (iv)

(v) na/feif. Let P = {{i : 21; ^ e} : e G £}. Then (JP = UJ by (iii) and (i), but P is finite by (ii) so, by (iv), there is some equation e = (to,*i) € S such that {i : 21; \j= e} £ T. Choose n e w s o that to, there are x\,...,xln £ Ai such that tf* (x\,..., x'n) / if ; (x\,..., xln) iff 21; ^= e. (Choose appropriately if the latter holds, otherwise choose arbitrarily.) It follows that {i : tf (x\,...,

x'n) / t f ; (x{,...,

xi)} G F,

but T is an ultrafilter, so (3.70)

{i : t*< (x{,..., xi) = tf ; (x\,...,

xi)} $ T.

11. ALGEBRAS OP SETS AND RELATIONS

For each k < n, let Xk {x\, : i £ a;). According to the definition of the direct product 1^21, w e have, for every t 6 Frn,

hence, by (3.70),

(3.71)

*? *(*!,. .,xn)/F

tP a (x!,..., xn)lT.

It can be shown by induction on terms that if t 6 Frn then tna/J Consequently, by (3.71),

This implies that 21/F ^= e, contrary to (v) and (i).

11. Algebras of sets and relations Let U be a set. Then 56 (U) is a set and 56 (U2), the powerset of U2, is also a set, called the set of binary relations on U. We are interested in algebras of relational type whose universes are 56 (U) and 56 (U2). The next theorem shows that the requisite operations are sets (which may be then paired with appropriate universes to form the desired algebras). Theorem 98. For every set U € V, the following sets exist and are functional (3.72)

Uu := {{{x, y), x U y) : x, y € Sb ([/)},

(3.73)

~u :={{x, U~ x) : x e Sb ([/)}, |t, : = { « * , y), x\y) : x,y e Sb ( [ / ) } ,

(3.74) (3.75)

~

lu

^ { ( x , ^ " 1 ) : x G 56 ([/)}.

P R O O F . Recall from T h . 49 t h a t t h e class {{{x,y} ,xUy) : x,y £ V} exists a n d is functional. Intersect t h i s (possibly p r o p e r ) class w i t h t h e set ( 5 6 (U))3 t o get t h e set Uu'-

Uc/ = { « * , ! / ) , x U y) : x, j , £ V} n (56 (t/)) 3 . For complementation with respect to U, first note that the difference operation {{{x,y) ,x~y) : x,y £ V} exists by Th. 49. The class {{y, {U,y)) : y £ V} exists because {{y, (U, y))-.yev} = CT 1 n v|{([/, t ^ l l P - 1 , so ~(7 may be obtained in this way: ~u = {(y,{U,y))

: y £ V}\{((x,y)

,x~y)

: x,y £ V} n ( 5 6 ( f / ) ) 3 .

For the rest, invoke Th. 49 and note that

\u = {({x,y),x\y):x,y€V}n(Sb(U))3, ~lu = {(x^-1)

: x £ V} n (56 {U)f. U

142

3. GENERAL ALGEBRA

In the next theorem we see that sometimes Sb (U) is closed under the operations defined in (3.72)-(3.75). Theorem 99. LetUeV

be any set.

(i) Sb (U) is closed under both Uu and ~u, and Uu : (Sb (U))2 -> 56 (U), ~u : Sb (U) -> Sb (U). (ii) If U is transitive, then Sb (U) is closed under \u and \u : (Sb (U)f -> Sb (U) . (iii) If U is symmetric, then Sb (U) is closed under ~lu

~lu and

: 56 (U) -> 56 (U).

PROOF. By Th. 98, Uu, ~u, \u, 1[7 are functional binary relations and also sets. By Th. 68, (56 (U)f is the domain of Uu and 56 (U) is the domain of ~u- By Th. 70, (56 (U))2 is the domain of \u and 56 (U) is the domain of " V If U is transitive, then (56 (U))2 is closed under |, for if x,y 6 56 ([/), then x \y Q U\U C U, hence x\y G 56 (U). If U is symmetric, then 56 (U) is closed under " \ for if x e Sb (U), then a;"1 C U'1 = U, hence a;"1 6 56 (IT). The closure conditions stated in the previous theorem have the following corollary. Theorem 100. Let U and E be arbitrary classes. (i) IfU is a set, then {Sb (U) ,Uu,~u) is an algebra of Boolean type. (ii) If E is a set and an equivalence relation, then (Sb (E) ,UE,-E,\E,~1E,ld

n E)

is an algebra of relational type. 11.1. Square relation algebras and equivalence relation algebras. We introduce names and notation for the algebras in Th. 100. For every set U, let <&l(U):=(Sb(U),Uuru), and refer to 93(([/) as the Boolean algebra of subsets of U. If E is both a set and an equivalence relation, then 6 b (E) is the relation algebra of subrelations of E, denned by &b(E)

:= (Sb (E) ,UE,-E,\E,~1E,M

r\E).

We refer to ©b (E) as an equivalence relation algebra. The class of equivalence relation algebras is {6b (E) :E~1\E = Ee\l}. Since every Cartesian square U2 is an equivalence relation, Th. 100 justifies the next choice of terminology, that for every set U,OTe(U) is the square relation algebra on U, defined by (3.76)

X

t

(

U

)

:

2

l

1

12. PROPER RELATION ALGEBRAS AND RRA

143

If U is empty then 9te (U) is an algebra with just one relation in it, namely 0. If U contains exactly one element, then ffte (£/) is an algebra with exactly two relations, namely 0 and U2, In both cases we have U1 = U2, Conversely, if U1 = U2 then U is either empty or has exactly one element. If U is finite then so is 9te (U). In particular, if U has exactly n elements, then the number of relations in SHe (U) is 2™ . For example, if U has exactly 4 elements, then *He (U) has 65,536 relations. If U has the same cardinality as the integers or the rationals, then 9te (17) has the same cardinality as the set of real numbers. Theorem 101 (Maddux [142, 1.7(4)]). Every square relation algebra is an equivalence relation algebra. In particular, for every set U, U"2 is an equivalence relation and SKe(i7) = 6 b (U2). The converse of the first part does not hold: not every equivalence relation algebra is a square relation algebra (or even isomorphic to one). In fact, 6 b (E) is not isomorphic to a square relation algebra whenever E is an equivalence relation with two or more equivalence classes. For example, if U and V are nonempty disjoint sets and E = U2 U V2, then &b (E) is an equivalence relation algebra which, according to Th. 105 below, is isomorphic to the direct product of the two square relation algebras 9te(E7) and 9te(V). The projection functions from &b (E) onto the two factor algebras are nontrivial homomorphisms since U and V are not empty. However, square relation algebras are simple and have no nontrivial homomorphisms. Consequently 6 b (E) is not isomorphic to any square relation algebra. 12. Proper relation algebras a n d RRA We say that SI is a proper relation algebra if SI is there is an equivalence relation B E V such that SI is a subalgebra of ©6 (E). Therefore, the class of proper relation algebras is (3.77)

S{eb(E);E~1\E = EeV}.

An algebra SI of relational type is said to be a representable relation algebra if it is isomorphic to a proper relation algebra. RRA is the class of representable relation algebras. (3.78)

RRA:=IS{6b(£):£T1|£ = .EeV}.

It is easy to see that the class of proper relation algebras is not closed under isomorphisms. Not every algebra that is isomorphic to a proper relation algebra is itself a proper relation algebra, simply because its elements may not be binary relations. Thus every proper relation algebra is representable, but not conversely. We say that p is a representation of SI over E if E is an equivalence relation, B g V , and p is an embedding of St into 6b (E). It follows that St 6 RRA iff there is a representation of SI over some equivalence relation E g V . We say that p is a square representation of 21 on U if p is a representation of SI over U2. A representation p : 21 —> &b (E) is said to be complete if it preserves all joins, that is, if X C ,4 and J2* x exists, then p(J2& x) = E S 6 ( B ) P*(x) = U/»*WA complete representation also preserves all meets.

144

3. G E N E R A L A L G E B R A

In order for an algebra 21 to be a subalgebra of (5 b (E), the identity element l'a of 21 must coincide with Id PI E, which is the identity element of (Sb (E). Hence l'a is an identity relation in the sense that V
(3.79) (3.80) (3.81)

g(a + b)=g(a)Ug(b), g(a) = E~g(a), g(a;b) = g(a)\g(b),

(3.82)

g{a) = (g(a))-\

(3.83)

g(V)D\dnE.

Let (3.84)

I:=g(V).

12. PROPER RELATION ALGEBRAS AND RRA

145

Then I is an equivalence relation. Define a function h from 21 into the proper relation algebra &b ({{r/I,s/I) : {r,s) E E}) by (3.85)

h := <{
Then h is a homomorphism from 21 into &b ({(r/I,s/I) an embedding whenever g is injective.

: (r,s) E E}), and h is

PROOF. First we note that (3.86)

5 (1)

= E

because 5(1)= 5(1'

+V)

(3.3)

= 3(1') U g(¥)

(3.79)

= IU(E~I)

(3.80), (3.84)

= E. Consequently 9(0) = 0 (3-3) (3.80)

9(0) =3(1) 1) = E^ E

(3.86)

— d) — v.

Next we show that / is an equivalence relation.

(3.87) / is symmetric because

I-1 = (g(V)y1

(3.84)

= 5(1')

(3.82)

= 5(1') = 7

assumption that 1' = 1' (3.84)

and 7 is transitive because

I\I = g(V)\g(V) = 9(V;V)

(3.84) (3.81) assumption that 1'; 1' = 1'

= 3(1') = 7

(3.84)

For every a E A we have g(a) = g(a)\I = I\g(a),

(3.88) because

g(a) = g(a;V) = g(V ;a)

assumption that a = a; 1'

146

3. GENERAL ALGEBRA

= g(a)\g(V) = g(V)\g(a)

(3.81)

= g(a)\I = I\g(a)

(3.84)

Let (3.89)

F:=/oE.

Then F is the function that maps each element r of Fd (E) to its equivalence class rjl. This function is defined on all sets, but produces a nonempty result only if it is applied to an element of Fd (E). Consequently, we have (3.90)

F-1\F={Fd{E)/I)\

F-1\E\F =

{(r/I,S/I):(r,s)eE}.

It follows from (3.85) and (3.89) that, for all a € A, (3.91)

= F- 1 | 5 (a)|F.

ft(a)

Equation (3.91) is an alternate definition of h that could replace (3.85) in the statement of the theorem. Next, note that F\F~1=I

(3.92) since 1

1

(3.89) 1

- ) (2.425) = I

(3.87)

Next we show that h is injective under the assumption that g is injective. Let us suppose that a,b €. A and h(a) = h(b). Then g(a)=g(a)nF\\/\F-1 1

= F|(F- |5(a)|F)|F-

(3.89), (2.420) 1

= F\h(a)\F-1 = F\h(b)\F~ 1

rot. (3.91)

1 1

(3.91) |-assoc

= I\g(b)\I

(3.92)

= g(b)

(3.88)

From the assumption that g is injective we conclude that a = b, completing the proof that h is also injective. What remains is to show, for all a,b € A, that (3.93)

h(a + b) = h(a)\Jh(b),

(3.94)

h(a) = {(r/I, s/I) : (r, s) e E} ~ h(a),

(3.95)

h(a;b) = h(a)\h(b),

(3.96)

h{a) = (h(a)y\

(3.97)

h(V) = (Fd(E)/I)\

12. PROPER RELATION ALGEBRAS AND RRA

147

What follows are the computational proofs of these claims. For (3.93), no properties of / are needed. (3.91) 1

=

(3.79)

F- \(g(a)Ug(b))\F 1

=

1

|-dist

F- \g(a)\FUF- \g(b)\F

(3.91)

= h(a) U h(b) For (3.94) it suffices to observe that h(a)nh(a) = F-1\g(a)\FnF-1\g(a)\F

(3.91) rot.

= F-'Ma)

|-assoc, (3.92)

n I\g(a)\I)\F

1

(3.88)

= F- \(g(a)ng(a))\F =

1

F- \(g(a)n(E~g(a)))\F

(3.80)

and

h{a) Uh(a) = F \g(a)\F UF \g(a)\F = F-1\(g(a)U 9(a))\F

(3.91) |-dist

= F- 1 |( 5 (a)U (E~g(a)))\F

(3.80)

1

= F~ \E\F = {(r/I,s/I): (r,s)£E}

(3.90)

Proof of (3.95): h(a;b.) = F- 1 | 5 (a;6)|F

(3.91)

1

= F- \(g(a)\g(b))\F 1

= F- |( 5 (a)|/|3(6))| F 1

1

= F- \g(a)\F\(F- \9(P)\F) = h(a)\h(b)

(3.81) (3.88) (3.92), |-assoc (3.91)

Proof of (3.96): h{a) = F-1\g{a)\F =

(3.91) 1

F-\g{a))- \F

= (Ha))

(3.82) (3.91)

148

3. GENERAL ALGEBRA

Proof of (3.97): ft(l') = F - 1 | f l ( l ' ) | F

(3.91)

1

= F- \I\F

(3.84)

C F-X\F

(3.89), (3.87), (2.219)

= (Fd(E)/I)

1

(3.90)

n 13. Closure of RRA under subalgebras a n d products Closure of RRA under I and S is easy to prove. Theorem 103. RRA = IRRA = SRRA. PROOF. Let K = {6b (E) : E'^E = E g V}. Then RRA = \SK by (3.78), IRRA = = \SK

(3.40)

= RRA, SRRA = SISK = \SSK

(3.44)

= \$K

(3.38)

= RRA.

If 21 G RRA then SI has at least one representation over some equivalence relation. Suppose we have system of representations. Consider an index set I 6 V and suppose that for each index i g I there is a representation pi ; A —> Sb (Ei) where Ei is a set and an equivalence relation. Define a function h : A —> Sb {[JieI Ei) by setting, for each a € A, h{a):=\Jpi{*)iei

It is easj to construct small examples which show that ft may not be a representation of 21. However, by the following theorem, it will be a representation if the fields of the relations Ei are disjoint. Theorem 104. Assume E is an I-indexed system of nonempty equivalence relations, that is, 0 ^ I g V, E : I —¥ V, and, for all i,j 6 I , i) n Fd{Ej) = 0 Then (Jiei -®« *

s an

equivalence relation and

13. CLOSURE OP RRA UNDER SUBALGEBRAS AND PRODUCTS

149

via the isomorphism ((RnEi-.iei):ReSb

(\Ji€IEt)).

Furthermore, if SI is a nontrivial algebra of relational type, p : I —> V, and, for all i € /, pi is a homomorphism from 21 into 6b (Ei), then (\JieI pi{a) : a 6 A) is a homomorphism from 21 into &b (Uiej Ei). Th. 104 is useful because any system of representations can be "isomorphically altered" by various set-theoretical devices so that the disjointness condition is satisfied. One case where this procedure is not needed occurs in the next theorem, by which an equivalence relation algebra may always be regarded, up to isomorphism, as the direct product of square relation algebras. Theorem 105 (Lyndon [133], Maddux [139, Th.8(8)(9)], [142, 1.7(5)]). // E £ V is an nonempty equivalence relation then U£Fd(E)/E

via the isomorphism ((RClU2 :Ue Fd (E)/E) :ReSb

(E)) .

Furthermore, for each U e Fd(E)/E the function (RnU2 : Re Sb (E)) is a homomorphism from &b (E) onto 9ie(U). Theorem 106. \{&b (E) : E'1^

= E £ V} = P{91e (U) : U € V}.

Let E'1^ = E e V. If 0 ^ E, use Th. 105 to conclude that 6b (E) e P{9k (U) : 0 ^ U e V}. If 0 = E, then 6b (0) = *Re (0) € P{9k (U) : U E V}. For the opposite inclusion, note that the direct product of a system of algebras in {$Re (U) : U € V} is isomorphic to another such product from which all factors of the form *He (0) have been deleted. The second product is in turn isomorphic to an equivalence relation algebra by Th. 104 and the remark after it. The key point here is that nonempty equivalence relations do not have any empty equivalence classes. PROOF.

Theorem 107. RRA = SP{*Re (U) :U eV} = PRRA. PROOF.

First, note that RRA = IS{6b (E) :E~1\E =--Ee V} = SI{6b (E) :E~1\E =--E€ V} = SP{me(U) .UeV}

(3.78) (3.44) Th. 106,

and then PRRA = PSP{«Re(I7) : U

ev}

CSPP{me(U) :U € V } = SP{mt{U) -.u e V} = RRA

(3.46) (3.45)

150

3. GENERAL ALGEBRA

C PRRA.

(3.45)

The primary goal of the next few sections is to prove the theorem of Tarski [232] that HRRA = RRA. For this we will need basic facts about homomorphisms, kernels, and congruence relations of representable relation algebras. After that we present some results concerning special kinds of relations in proper relation algebras. From the previous sections we see that it is fairly easy to show that RRA is closed under the formation of subalgebras and direct products of systems of algebras. It is, however,somewhat harder to show that RRA is closed under homomorphisms, that is, RRA contains all it quotient algebras. The first proof that HRRA = RRA is due to Tarski [232] in 1955. Combined with SRRA = PRRA = RRA, this is enough to conclude, on the basis of BirkhofPs Th. 95, that RRA has an equational axiomatization. In 1964 Monk [176] proved that no equational (and, in fact, no first-order) axiomatization of RRA is finite. Monk's proof utilizes algebras that arise from projective geometries according to the 1961 construction of Lyndon [135], From the results of Lyndon [135], which connect projective lines and planes to relation algebras and their representations, respectively, together with the 1949 theorem of Bruck-Ryser [45] on the nonexistence of projective planes, it follows that there are infinitely many projective geometries, each consisting of a single line with a finite number of points, whose corresponding algebra of relational type is not in RRA. Furthermore, it is quite straightforward to show that any nonprincipal ultraproduct of these algebras is in RRA. It follows from this observation by general model-theoretic considerations that RRA does not have any finite first-order axiomatization, let alone an equational one. In 1965 Monk [177] converted the algebras of relational type derived from projective lines into cylindric algebras, and, by essentially the same argument involving ultraproducts, showed that there is no finite equational (or even firstorder) axiomatization for the class RCA3 of representable 3-dimensional cylindric algebra. In 1969 Monk [180] went on to prove that there is no finite equational (or first-order) axiomatization for RCAa whenever 3 < a < u. To deal with the dimensions beyond 3 he invented new cylindric algebras whose nonrepresentability is based on Ramsey's Theorem; see, e.g., Graham-Rothschild-Spencer [85]. These cylindric algebras, converted back into algebras of relational type, provide another proof that RRA has no finite equational axiomatization. Indeed, they are the ones used in the proof of Monk's Th. 458 below. 14. Relational ideals Recall that an algebra is in RRA iff it is isomorphic to a subalgebra of &b (E) for some equivalence relation E. Therefore, to study homomorphisms on representable relation algebras, it suffices to consider homomorphisms on subalgebras of equivalence relation algebras. Let us suppose that E is an arbitrary equivalence

14. RELATIONAL IDEALS

homomorphism h

congruence relation h\h-L

relational ideal {R : h(R) = ft(0)}

( C o E ) n ( i x V)

C

0/C

{{R,S):ReSeI}

I

TABLE 2. Homomorphisms, congruences, ideals

relation, and that 21 is a subalgebra of ©6 (E). Note that E is actually determined by 21 since E is the Boolean unit of 21, that is, 1st is an equivalence relation, but we choose to call it E (a more suggestive notation). We will define the notion of relational ideal and prove theorems which say, in aggregate, that homomorphisms, congruence relations, and relational ideals are interchangeable in the sense that each determines one of the others by some particular construction, and that these constructions are inverses of each other. Table 2 contains a summary. The first line asserts that an arbitrary homomorphism h 6 HoSl on a proper relation algebra St C SHe (E) gives rise (according to Th. 85) to the congruence relation h\h-1 £ Co%, which appears in the second column of the table, and h also produces (according to Th. 110 below) the relational ideal {R ; h(R) = fc(0)}, called the kernel of h. In the second line, an arbitrary congruence relation C E CoSt on the algebra 21 gives rise to the relational ideal 0/C and (according to Th. 84) to the homomorphism (C o E) n (A x V). Finally, an arbitrary relational ideal I on SI gives rise (according to Th. I l l below) to a congruence relation {{R, S) : R,S G A,RQ S G I}. The homomorphism corresponding to the relational ideal I is : R,S

S e I}oE) n(A xV).

From every homomorphism h we get a congruence relation ftlft"1 by Th. 85. Furthermore, every congruence relation arises in this way, i.e., every congruence relation C is equal to ft|fe~x for some homomorphism h by Th. 84. Thus the function ^ft|fe-1 : ft € i?o2l) maps Ha3l onto CoSt. Conversely, every congruence relation C 6 Co% gives us a homomorphism, in fact, a homomorphism from the algebra 21 onto the quotient algebra 21/C. However, not every homomorphism arises this way. The ones that do are homomorphisms that we may call quotient homomorphisms —homomorphisms onto quotient algebras. Every homomorphism corresponds to a quotient homomorphism by Th. 85. The image of 21 under a homomorphism ft is isomorphic to the quotient of SI with respect to the congruence relation h\h~x. Thus we may write

Since not every homomorphism arises from a congruence relation, the fact that the processes leading from one to the other are inverses of each other must therefore be expressed in terms of congruence relations. Specifically, we see from Th. 84 and Th. 85 that every congruence relation C is equal to the congruence

152

3. GENERAL ALGEBRA

relation ((Co E) n (A x V ) ) | ( ( C o E ) ( l ( i x V))- 1 , which is determined by the homomorphism (CoE)n(AxV), which is in turn determined by the congruence relation C. We say that I is a relational ideal of the proper relation algebra St C 6 b (E) if I has the following properties: (i) ICA, (ii) if Re A and RCS el then Re I, (iii) if ii, S 6 I then RUS el, (iv) if ii 6 I then E\R g i, ii|B € i, and ii" 1 g I. A relational ideal I of 21 is proper if I ^ A. I is a maximal relational ideal of SI if I is a relational ideal of St and I = J whenever J is a relational ideal of SI such that / C J. A couple of useful observations are made in the next theorem. Theorem 108. Suppose I is a relational ideal of a proper relation algebra SI and E = 1^. For every R e A, R e I & E\R\E el. I is proper iff E <£I. PROOF. If i i g I, then E\R g I since I is closed by relative multiplication on the left by E, and then E\R\E g I since I is closed by relative multiplication on the right by E. For the converse, first note that ii C RIR'^RKR-^R) 1

1

(2.176), |-assoc

C.E|.E~ |.R|(.E~ |.E)

RCE

= E\R\E

E'1=E

= E\E

If E\R\E e I then Re I because / is closed under subrelations. If J is proper then I = A, hence E = l a € A = I. For the converse, suppose Eel. Then A = I since, for every Re A, we have RC E e I, hence R e I since I is closed under subrelations. Thus I is proper whenever Eel. A relational ideal I of 21 may not be an ideal on the set E. It may not contain all the subrelations of any given Re I, simply because those subrelations may not all be elements of the subuniverse A of 6 b (E). Suppose, however, that 21 = &b (E). Then A = Sb (E) and every relational ideal of 21 is also an ideal on the set E because of conditions (i)-(iii) above. On the other hand, an ideal I on E is not a relational ideal of 6 b (E) if condition (iv) fails and / is not closed under conversion or relative multiplication by E. For example, if 0 ^ E = U2 then there are exactly two relational ideals of 6 b (E), one is 0 and the other is 56 (E). Nevertheless, there is a correspondence between relational ideals and ideals on sets, which is expressed in the following theorem. Theorem 109. Let E eV be an equivalence relation. If I is a relational ideal of&b (E), then {Fd (R)/E :R el} is an ideal on the set Fd (E)/E of equivalence classes of E. If I is an ideal on Fd{E)/E then [){Sb (U[/ejr U2) : X e 1} is a relational ideal of&b (E).

14. RELATIONAL IDEALS

14.1. Kernel of a homomorphism. If h is a homomorphism on a proper relation algebra 21, then the kernel of h is the set of relations in A denned by ker(h) := {R : h(R) = l

Since h\h~ is an equivalence relation (actually a congruence relation by Th. 85), ker(h) is the equivalence class of 0 with respect to h\h~x. 14.2. Kernel is a relational ideal. In the next theorem we see that the kernel of a homomorphism h on a proper relation algebra is a relational ideal. Furthermore, the kernel determines the congruence relation h\h~l by a formula which, according to Th. I l l below, will produce, when applied to an arbitrary relational ideal 7, a congruence relation whose correlated quotient homomorphism has I as its kernel. Theorem 110. Assume 21 is a proper relation algebra, E = l a , and h G if om (21, 23) is a homomorphism from 21 into an algebra 93 = {B, +,~, ;, ", 1') of relational type. Then (i) ker(h) is a nonempty relational ideal of %, (ii) if 93 is nontrivial (has more than one element) and h is onto, then h(E) ^ /i(0), E £ ker(h), and ker(h) is proper. (iii) hlh'1 = {(R, S) : R,S G A, Re S G ker(h)}. PROOF. Proof of (i): First note that 0 E ker(h) since 7i(0) = 7i(0), so ker(h) is nonempty. Next we show ker(h) is closed under subrelations. If S G A and S C R e ker(h) then = h(Sr\R)

so S G ker(h). Then

SCR

= h(S) h(R)

h G Hom{% 93)

= h(S) h{
R € ker(h)

= h(Sn
hE

To show ker(h) is closed under unions, assume R, S G ker(h).

h(RL)S) = h{R) + h(S)

h e Horn (21, 23) i?,S'Gfcer(/i) ft 6 Horn (21, 23)

so R U 5* e ker(h). li R E ker(h), then iZ"1 6 ker(h) since -1

) = (fc(fl)r

ft

effom (21, 23) i? G fcer(/i) fte Horn (2t, 23)

154

3. GENERAL ALGEBRA

R\E G ker(h) since h(R\E) = h{R);h{E)

ft

= h(®);h(E)

<E #om(2l, 58) Reker(h)

= h(Q\E)

h eHom(21,58)

and, similarly, S|iZ G ker(h). Thus ker(h) is a nonempty relational ideal. Proof of (ii): It follows from closure under subrelations that h(E) ^ /i(0), for if h(E) = /i(0) then h(R) =ft(0)whenever i? D R € J4, hence the image of A under ft contains exactly one element, contrary to the assumption that 58 is nontrivial and h is onto. Proof of (iii): If (R, S) G h\h~l then R, S G A and h(R) = h(S), so ) = h((RnS)u(EnS)) = h(R) h(S) + h(R)

(2.6) h(S)

= h(R) h(R) + h(R) h(R)

h(R) = h(S)

= h((Rn~R)\j(Rr\R))

ft

hence RQS G ker(h). For the converse, suppose RQS G ker(h). Then h(RQS) = ft(0), so = h(RU0) h{%)

(2.62) he Horn (21,58)

h{RQS) = h(RU(RQS)) )

(2.82),

and, similarly, ft(S') = /i(fl U 5). It follows that fc(i?) = fc(S), hence (i?, 5) G The kernel of a homomorphism may fail to be proper even if the target algebra is nontrivial. For example, if 58 is a nontrivial algebra in which {0<s} is a subalgebra of 58 and h is a homomorphism from a proper relation algebra 21 into 58 such that h(R) = 0
14. RELATIONAL IDEALS

158

reflexive over A, for if R £ A then R 9 R = 0 G I, hence (R, R) £ C. Next, C is symmetric because RQS = SQRby (2.74). Finally, to prove C is transitive, assume {Q, R) g C and {#, S) g C. Then Q 8 -R 6 I and R Q S 6 J, so (Q Q R)U (RQ S) £ I since / is closed under unions. But then Q Q S G / since Q 9 S C (Q e -R) U (ij 8 S) by (2.84) and I is closed under subrelations. Hence (Q, S) 6 C. Therefore C is an equivalence relation. Assume {R, S) g C. Then # e S € I, but R~QS = ReS by(2.83), so 'RQS € J and (R, S) 6 C Also, (i?0S)~ £ J since J is closed under conversion, but R-1 9 S " 1 = ( B O S ) " 1 by (2.142), so (R'1^'1) g C. Thus C is a congruence relation with respect to the unary operations of complementation and conversion. Assume {P,Q) £ C and (R,S) G C. Then P Q Q £ I and Re S £ I, so (PQQ)U(RQS) g J since J is closed under unions. But (PUR)Q(Ql)S) C (PQ Q)U(RQS) by (2.85), so (PUR)Q(QUS) € J since J is closed under subrelations. Consequently, {P U R, Q U S) G C This shows that C is a congruence relation with respect to union. Next we deal with relative multiplication. From PQQ 6 J, R Q S £ I, and the closure of / under subrelations it follows that p n Q, R n Is, Q n P, S n ~R e J. Let £ = la. Because / is closed under relative multiplication by 13, we conclude that

(3.98)

(P n Q)\E, E\(R n 5), (Q n P)|s, s|(5 n 1) e /.

Next we use (3.98) and the closure of I under unions to show that two more relations are in I. C(PC\Q)\R U

P|(#ns)

c(PnQ)\E u B|(flnS) £l

P\RnQ\S=(P1[R)nQ\S c(QnP)\S u 0|(5nS) C (Q(~\P~)\E U E\(Sr\R) el

(2.130) |-mon (3.98)

(2.130) |-mon f3.98)

Since the union of two relations in I is also in / , we have

p\RQQ\s = p\Rn~Q\s u T\R~nQ\sei, hence (P\R, Q\S) 6 C, completing the proof that C is a congruence relation. Proof of (ii): If R E 0/C, then (R, 0) G C, but R = ii 9 0 by (2.77), so .R e /. Conversely, if R g I, then i? 8 0 = -R 6 J, hence {J2,0) 6 C and ii 6 0/C. Proof of (iii): We know that the domain of C is A, since, by (2.75), R Q R = 0 for every i? € A. Thus Do {C) = A and i J C i x V . Let A be the homomorphism determined by C, that is, h := (Co E) fl (A x V). By Th. 84, h is a homomorphism from SI onto the quotient algebra 21/0. Each element R £ A is mapped by h to

156

3. GENERAL ALGEBRA

its C-equivalence class R/C; see (2.395). To complete the proof we must show that the kernel of h is I. Observe that the following statements are equivalent for every R E A: R £ ker(h), h(R) = ft(0), R/C = 0/C, V S (S e i ^

({R, S) £C

vs(5 eA => {Res VS(SG A => (Res

<S>

(0, S) £ C)),

el <* 0 eS ei

o-

el)),

Sei)).

Suppose first that R £ ker(h). Then the last statement holds. Applied with S = R, it yields Re R € I <S> R 6 7. But Re R = 0 and 0 6 7, so R 6 7. For the converse, suppose R £ I. To show R £ ker(h) it suffices to prove, for any S £ A, that fleSe/» 5 £ /. I f S e / then R U S £ I by closure of / under unions, but Re S C RU S, so Re S E I, and, conversely, ii Re S E I then 5 ' C i ? U 5 ' = i?U(i?eS') G / b y (2.82) and closure of 7 under unions, so S £1. 15. S and H commute on proper relation algebras Theorem 112. Assume E £ V is an equivalence relation. Then (3.99)

HS{6b (E)} = SH{6b (E)}.

PROOF. We may assume E is not empty, since &b (0) is a 1-element algebra of relational type, so S{6b (0)} = {6b (0)} and HS{6b (0)} = H{6b (0)}. But H{6b (0)} is the class of 1-element algebras of relational type and this class is closed under subalgebras, so SH{6b (0)} = H{6b (0)} = HS{6b (0)}. We have HS{6b (E)} D SH{6b (E)} by (3.43), so assume 35 <E HS{6b (£)}. Then there is a subalgebra 21 C 6 b (E) and a homomorphism h E Horn (01, 23) from 21 onto 93. Let

(3.100) J :={R:RCE,RC\£ker(h)}. Next we show that J is a relational ideal of 6 b (E). Suppose R,Se,J. Then there are R', S" such that R C R' £ ker(h) and S C 5" £ ker(h). We have RUS C R'US' and R' U S' £ ker(h) since ker(h) is closed under unions, so R U S £ J. Thus J is closed under unions. For closure under subrelations, suppose R C S £ J. Then S C S' £ ker(h) for some 5", but then R C S' £ ker(h), hence R £ J. Finally, for closure under conversion and relative multiplication by E, suppose R £ J. Then R C S 6 ker(h) for some S. Since ker(h) is a relational ideal on 21, we have S^1 £ ker(h), S\E £ ker(h), and E\S £ ker(h), but by monotonicity we also have R'1 C S~\ R\E C S\E, and £|7Z C E\S, so i?" 1 G J, ij|£ G J, and

E\Re J. Let C := {{R, S) :R,SCE,ReS

/ : = (C o E) n (Sb (E) x V),

e,7],

IB. S AND H COMMUTE ON PROPER RELATION ALGEBRAS

We then know that C G Co&b (E) by Th. I l l , / is a homomorphism from 6b (E) onto 6b (E) /C by Th. 84, J = ker(/) by Th. I l l , and C = / I / " 1 by Th. llO(iii). To complete the proof we will show that g is an isomorphism from 23 into 6b (E)/C, for then 23 G IS{6b (E) /C) C ISH{6b (E)} = SIH{6b (E)} = SH{6b (E)}. Here is the situation: 6b (E)

1/ 23 — 9 —^ 6b (E) /C To show g is functional, first note that /i|ft-1 C f\f~1, for if {R, S) € h\h-1, then fl,SeA and h(R) = h(S), so Re S e ker(h) C J, hence (i?, S) G C = / I / " 1 . Then, since / is functional, we have

g-'b = r ' l W O l / c r ' K / i r 1 ) ! / = (/- L I/)I(/- 1 I/) c id|id = id. To show g is injective, assume (b,c) G g\g-1 Then (&,c) G ^ ~ 1 | ( / | / ~ 1 ) | ^ ! so there are ii, S € A such that 6 = ft(B), (R, S) € / I / " 1 , and h(S) = c. Then (fl, S) € C since / I / " 1 = C so fle5 G J. By (3.100), ReS C|e ker{h), but fcer(/i) is closed under subrelations and i? 9 S £ A, so i? 9 5 6 ker(h), hence /i(i?) = /i(S'), i.e., 6 = c. We only show that g has the homomorphism property with respect to the operation +sg- Computations for the other operations of 23 are similar. Let 6, c € B. Since ft is a homomorphism onto 23 we may choose R, S € A such that h(R) = b and h(S) = c. Then (6, R) G /i" 1 and (c, 5) eh'1, while (fl,/(fl)) G / and (S,f(S)) G / , so (b,f(R)) G ft"1!/ = 5 and {c,f(S)) G ^ J | / = S) *-e-) sC7) = f(R) a n d S(c) = /(-S*)- The operation corresponding to +sg in the quotient algebra 6b (E) /C is Lt/c- What we wish to show is gib + c) = g(b) \J/Q g(c). Since h and / are homomorphisms, we have h{R US) = h{R) + h{S) =b + c, i.e., {b + c,RUS} G ft" , and 1

f(RUS) = f(R)U/cf(S), i.e., {R U 5, /(i?) U/ c /(5)) e / , hence <6 + c, /(i?) U/ c f(S)) e ft"1!/ = S, i.e., c) = f(R) U/c f(S) = g(b) U/c g(c), as desired. Equation (3.100) shows how to extend a relational ideal of a subalgebra to a relational ideal of the whole algebra. The next theorem shows how to extend a relational ideal to a larger relational ideal within the same algebra. It is used as a lemma twice in the subsequent sections. Theorem 113. Assume 21 is a proper relation algebra, I is a proper relational ideal o/2l, and R G A. Let E = 1<% and J:={S:SeA, 3T(T G / A 5 C E\R\E U T)}.

158

3. GENERAL ALGEBRA

Then J is a relational ideal. If E\R\E (/L I then J strictly contains I. PROOF. It is very easy to show that I C J and E\R\E € J, so J strictly contains / whenever E\R\E £ I. If Q,S € J then there are T,U € / such that Q C E\R\E U T and 5 C E\R\E U U, hence QUSC

(E\R\E U T ) U (S|i?|S U U) = E\R\E U (T U C/),

but T U t / G l , s o Q U S G J. Thus J is closed under unions. Suppose 5 G J, so there is some Tel such that 5 C E\R\E U T. Then T " 1 G / and E\R\E = l E\R~ \E since we have E\R\E <1 E\{R\R~l\R)\E T

(2.174)

1

C (JE |JE7)|JR- |(£?|i7)

|-assoc,

l

= E\R~ \E and, similarly, EIR'^E

RCE

E = E\E,

C S|i?|S, so, since S" 1 = £,

5" 1 C {E\R\E U T)" 1 = S " 1 ^ " 1 ^ " 1 U T" 1 = S|fl|£; U T~\ which establishes that S" 1 € J. To see that E\S £ J, observe that E\T £ I and E\S C S|(S|i?|S U T) = S|S|i?|S U £|T = S|i?|S U E\T. Similarly, S\E € J.

D 16. Peircean ideals

Let 21 be a proper relation algebra and E = la- We say that 7 is a Peircean ideal of 21 if I is a relational ideal of 21 and, for every R £ A, (3.101)

£ | f l | £ ! £ / *>

E~E\R\EeI.

By Th. 108, (3.101) may be replaced in this definition by According to the next theorem, if the kernel of a homomorphism h on Sb (i?) is a Peircean ideal then the image of Sb (E) under h is an algebra of relational type that has Peirce's remarkable property, namely, that l;x;l is either 0 or 1, for all x. Theorem 114. Assume h is a homomorphism from a proper relation algebra 21 onto a nontrivial algebra 03 = {B,+,~, ;, ",1') of relational type and ker(h) is a Peircean ideal of 21. Then, for all R G A, h(E\R\E) / ft(0) *> h{E\R\E) = h(E). PROOF. We know that ft(0) # h(E) by Th. llO(iii), so if h{E\R\E) = h(E) then h(E\R\E) / ft(0). For the converse, suppose h(E\R\E) / ft(0). Then E\R\E $ ker(h), but ker(h) is a Peircean ideal, so E~E\R\E e feer(/i), hence h(E~E\R\E) = h(®). Then E

17. CLOSURE OF RRA UNDER HOMOMORPHISMS

h(E\R\E) h(E\R\E U 0)

h£Hom(Vl,
Theorem 115. Let % be a proper relation algebra. I is a maximal proper relational ideal of % iff I is a Peireean ideal of%l. PROOF. Let E = l a . Then 21 C 6b (E). Assume / is a Peireean ideal. Since 21 is an algebra, it is not empty, so let R £ A. Then either E\R\E € I (so I is nonempty) and E ~ E\R\E £ I (so I is proper), or the other way around. Either way, I must be proper and nonempty. Now we show / is maximal by supposing, to the contrary, that / is not maximal and deriving a contradiction. Assume there is some strictly larger proper relational ideal J D I and choose a relation R £ J ~ /. From R £ J we get E\R\E e J since J is a relational ideal. From R ^ / we conclude, since I is Peireean, that E ~ E\R\E £ I. But I C J, so we now have both E ~ E\R\E £ J and E\R\E £ J. Since J is a relational ideal it is closed under unions. Consequently, E = (E~E\R\E) U E\R\E 6 J, contradicting the assumption that J is proper. For the converse, assume that / is a maximal proper relational ideal. We will derive a contradiction by assuming that I is not Peireean. Accordingly, suppose there is some R £ A such that either E\R\E £ I and E ~ E\R\E £ I or else E\R\E £ I and E ~ E\R\E £ I. The former case implies E = E\R\E U (i5~ E\R\E) 6 /, contradicting the assumption that I is proper, so E\R\E ^ I and E~E\R\E £ I. Let J := {S : S £ A,3T(T £ I A S C E\R\E U T)}. By Th. 113, J is a relational ideal strictly containing /. From our assumption that / is a maximal proper relational ideal we conclude that J = A. In particular, E~E\R\E £ J, so there is some T £ I such that E ~ E\R\E C E\R\E U T. This last equation implies E C T, so E £ I since / is closed under subrelations, contrary to our assumption that / is proper.

Theorem 116. Every proper relational ideal I of a proper relation algebra 21 is contained in a maximal proper relational ideal of%l. PROOF. The set of proper relational ideals of 21 that contain /, partially ordered by inclusion, is a poset that is closed under unions of chains. By Zorn's Lemma it has a maximal element.

17. Closure of RRA under homomorphisms For every B e V , let (3.102)

P s : = { p : p C £ = E\p\E, p\E\p C Id}.

Theorem 117. Let E € V be a nonempty equivalence relation. (i) p £ Pg iff p C E and for every equivalence class U £ Fd{E)/E, there is some x £ U such that pDU2 = {{x, x}}.

160

3. GENERAL ALGEBRA

(ii) For every p £ PE, p\E\p = p = p " 1 C Id. (iii) / / R,S C E and p,q e P B , then (3.103)

E\p\R\q\E n E\p\S\q\E = E\p\(Rn S)\q\E,

(3.104)

E\R\p\E n E\p\S\E = £|.R|p|S'|.E,

(3.105)

E\p\R\q\E = E\q\R-1\p\E.

(iv) For ewen/ i J C E there are p,q€PE such that E\R\E = E\p\R\q\E. (v) IfR,SCE then there is some pEPE such that E\R\S\E = E\R\p\S\E. PROOF. Proof of (i): Assume p £ PE and U £ Fd(E)/E. Since U ^ 0, we may choose some u E U. Then {u, u) E U2 C i? = i?|p|i?, so there are x,j/ such that (K, a;) £ i5, (x,j/) £ p, and (y, u) £ i5. Prom (u, x) € E and (y,u) £ E we have x,y € U. Prom (a;,?/) £ p and p Q E = E~x we conclude that (x, y) E p\p 1\p C p|£7|p C Id, hence x = y. Thus {(a:, a;)} C U2 C\ p. For the opposite inclusion, suppose that v,w £ U and (t), w) £ p. Then, from (v,w) £ p, {w,x) E E, and {x,x) E p we have («,a;) 6 p|i?|p C Id, so v = x, and from {x,x)

E p, {x,v)

E E, a n d {v, w) E p we h a v e (x,w)

E p\E\p

C Id, so x = w. So

far we have shown that every p £ P ^ has the required form. It is easy to verify that any relation with that form is in P # . The remaining parts can be proved quite readily, either by making use of part (i) or by simple calculations, as shown by the following examples. Proof of (3.105): Since E'1 = E, q = q'1, p = p " \ and E\p\R\q C E, we have E\p\R\q\E C E:|p|.R|g|(g-1|i?-1|p-1|.Er1|.E: |~l E)

rot.

CElqlR-'lplE, and taking converses of both sides yields the opposite inclusion. Proof of (v). Let R, S C E. For every U E Fd(E)/E, if R\S H U2 = 0, let xu be any element of U, and if R\S fl U2 ^ 0, let xu be any element of Do(Sr\R~1\U2). Let p = {(xu,xu) : U £ Fd(E)/E}. Then p £ PE by part (i) and E\R\S\E = E\R\p\S\E. D The kernel of a homomorphism from Sb (E) onto a nontrivial algebra 03 of relational type determines a proper relational ideal ker(h) of 6b (E). The set of maximal proper relational ideals extending ker(h) corresponds to a system of homomorphisms on 23 that produces a subdirect decomposition of 23 into simple algebras. Calculations in the next theorem show that each such simple homomorphic image of 23 is isomorphic to a square relation algebra. Any subdirect product of square relation algebras is representable, so it follows that 23 £ RRA. The representability of the homomorphic images of &b (E) leads directly to Th. 120 below, that RRA is closed under homomorphisms. Various proofs of this fact can be found in the literature, for example, Tarski [232], Maddux [138], Jonsson [113, 115], and Hirsch-Hodkinson [99]. Theorem 118. Assume E £ V', E is a nonempty equivalence relation, and h is a homomorphism from &b (E) onto a nontrivial algebra 23 = {B,+,~, ;, ", 1')

17. CLOSURE OF RRA UNDER HOMOMORPHISMS

of relational type. For every R C E, let a (R) := {(p, q):p,qePE,

h(E) = h(E\p\R\q\E)}.

Then, for all R, S C E, (3.106)


(3.107)

a(E) = (PE)2,

(3.108)

tfRCSCEthena(R)Ca(S),

(3.109)

if ker(h) is a Peircean ideal, then a (R U S) = a (R) U a (S),

(3.110)

o-(E~R)r\o-(R)

(3.111)

if ker(h) is a Peircean ideal, then a (E ~ R) = ( P B ) 2 ~ a (R),

(3.112)

a(R\S) = a (R) \a (S) ,

(3.113)

a(R-1)=(a(R)y1,

(3.114)

cr (Id HE) D Id l~l ( P B ) 2 ,

(3.115)

ifker(h)

= %,

is a Peircean ideal then 03 = | C me (PE/cr

(Id n S ) ) .

PROOF. Proof of (3.106): If p,g <= P B then h(E\p\0\q\E) = ft(0), butft(0)^ by Th. 110(ii), hence fe(.E|p|0|c/|.E) ^ ft(S), i.e., (p,g) ^ cr(0). This shows that

a(0) = 0. Proof of (3.107): If p,q 6 P B then £ | p | £ = E = E\q\E, hence E = E\E = (E\p\E)\(E\q\E) = E\p\E\q\E. This implies (p,q) ea(E). Proof of (3.108): Assume R C S C E and (p,q) G a(R). Then h(E) = h(E\p\R\q\E), so (p,g) E a (S) since = h(E\p\S\q\E) -h(E) = h(E\p\S\q\E)

h € Hom(2L,<8)

h(E\p\R\q\E)

= h(E\p\S\q\E n E\p\R\q\E)

h G Hom{% OS) (3.103)

Proof of (3.109): Assume p, q E PE, R, S C E, ker(h) is Peircean, and (p,q) E o-(RUS). Then h(E) = h(E\p\(RU S)\q\E) = h(E\p\R\q\E) + h(E\p\S\q\E)

|-dist, h E Hom(% «B)

By Th. 114, we have {h(E\p\R\q\E),h(E\p\S\q\E)} C {ft(0), h(S)}. If ft(S|p|7l|g|S) = h(E\p\S\q\E) =ft(0)then h(E) =ft(0)+ /i(0) =ft(0U 0) = ), contradicting h(E) ^ ft(0). Therefore either h(E\p\R\q\E) = h(E) or = h(E). In the former case, we have {p,q} E o~{R), and in the

3. GENERAL ALGEBRA

latter case, (p, g) G a (5). In either case, (p, g) G a (R) U a (5). This shows that a (R U S) C a (R) U cr (S). The opposite inclusion follows from (3.108). Proof of (3.110): Let p, g <E PE and i? C £ . If
= h{E)-h{E)

(3.103)

contradicting h(%) / /i(S). Therefore a (E ~ i?) n a (R) = 0. Proof of (3.111): (3.111) follows from (3.109) and (3.110). Proof of (3.112): Suppose {p, q) € a (R) \a(S). Then there is some r €. P B such that (p, r) G
he Horn (21, 35) h € Hom(% 03) (3.104)

From i?|r|S C i?|ld|S = R\S we get a(R\r\S) C
17. CLOSURE OF RRA UNDER HOMOMORPHISMS

163

Proof of (3.115): Assume ker(h) is Peircean. Let g = h-1\a. Note that g C B x ( P B ) 2 . Since h maps 56 (E) onto B and the domain of a is 56 (E), it follows that the domain of g is B. Next, we show that g is functional. Suppose (6,1Z) G g and (6,5) € g, where 6 € -B and K , 5 C (P B ) 2 . By the definition of g there are R,S C E such that /i(fl) = 6 = /i(S'), a (R) = 11, and a (5) = S. But then, for all p,q £ P.E, we have

= h(E\p);h(R);h(q\E) = h(E\p);h(S);h(q\E) = h{E\p\S\q\E), which implies that a(R) = cr(S), i.e., 1Z = S. Thus g is functional, and to compute g(b) it suffices to choose any R C E with h(R) = b and get g(b) = a(R). So far we have shown that g : B -¥ Sb ((Ps) 2 ) and (3.116)

g(h(R)) = a(R)

for every R C E. To see that g is injective, suppose that b,c £ B and 6 / c. Since ft is onto, we may choose R, S C E such that h(R) = b and ft(S) = c. Then ft(ii) # h(S), i.e., (R,S) i hlh'1. By Th. llO(iii), R Q S <£ ker(h). By Th. 108, E\(R Q S)\E <£ ker{h) and h(E\(RQS)\E) / ft(0). By Th. 114, since ker{h) is a Peircean ideal, we have h(E\(R 0 5)|£) = /i(£). By Th. 117 we may choose p,q € P B such that E\p\(R Q S)\q\E = E\(RQ S)\E. From the last two equations we get h(E\p\(RQ S)\q\E) = h(E), so (p,q) ea(RQS) = a(R)Qa(S)

(3.109), (3.111)

= g(h(R))eg(h(S)) = g(b)eg(c) so g(b) / g(c). Next we show that the following statements hold for all b, c €. B.

c)=g(b)Ug(c), =

(PE)2~g(b),

g(b;c) = g(b)\g(c),

Choose R, S C E so that h(R) = b and h(S) = c. Then, by (3.116), a (R) = g{b) and a (5) = g(c), so Choose R, S C E so that /i(fl) = 6 and ft(S') = c. Then, by(3.116), cr (i?) = g(b) and tr (5) = g(c), so

= g(h(RUS)) = cr(i?U5)

ft

G .Horn(21, 23) (3.116)

3. GENERAL ALGEBRA

= a (R) U a (S) = g(b)Ug(c),

(3.109)

g(b) = g(h(R)) = g(h(E~R))

ft effom (21, 35)

=

a(E~R)

(3.116)

= (PEf~a(R)

(3.111)

g(b;c) = g(h(R);h(S)) = g(h(R\S))

h € Horn (21, 23)

= a(R\S)

(3.116)

=

(3.112)

a(R)\a(S)

= g(b)\g(c), g(b) = g((h(Rjy) = g(h(R-1)) =

1

o(R- )

he Hom(%T,) (3.116) (3.113)

g(V) = g{h{\d n E))

h e .Horn(21, 23)

= <7 (Id HE)

(3.116)

2

(3.114)

3ldn(PB)

We may now apply Th. 102 (with E in that theorem taken to be (PE)2) conclude that

and

® =|C 6b ({(p/g(V),q/g(V)) : (p,q) e (P £ ) 2 }) via the embedding (3.117)

h := {{(p/g(V),q/g(V)) : (p,q) G g(b)} : b € B).

However, note that {(P/I,q/I)

(P,q) G ( P E ) 2 } = PE/g(V)

and g(V) = a (Id n S), so we also have Q3^|CfRe(P B /a(ld HE)). Thus © is isomorphic to a subalgebra of a square relation algebra whenever the kernel of h is Peircean. If ft is a homomorphism from 6 b (E) onto 25, then the kernel of h may not be Peircean and 2} may not be embeddable in a square relation algebra. Nevertheless, we can use (3.115) and the Assembly Lemma to show that 2} is representable.

17. CLOSURE OF RRA UNDER HOMOMORPHISMS

165

Theorem 119. Assume E € V, E is an equivalence relation, and h is a homomorphism from 6b (E) onto an algebra 03 = {B,+,~, ;, ", 1'} of relational type. Then 03 G RRA. PROOF. If 03 is trivial, then 03 ^ 6b (0) = $Re (0) € RRA. We therefore assume that 03 has at least two elements. By Th. 110(ii), ker(h) is proper. Let 6 and c be distinct elements of 03. Since h is onto, we may choose R, S C E such that h(R) = b and h(S) = c. Then h(R) / h(S), so ReS $ ker(h) by Th. 110(iii), hence E\(R Q S)\E $ ker(h) by Th. 108. Let (3.118)

Q:=E~E\(RQS)\E,

(3.119)

J :={S :SeA, 3T(T € kerih) A S C E\Q\E U T)}.

By Th. 113, J is a relational ideal. If J were not proper, we would have E\(RQ S)\E € J, hence, for some T € ker(h), (3.120)

E\(ReS)\ECE\Q\EUT.

But

E\{ReS)\EnE\Q\E c sKQns-^sKiieS)!^)!^" 1 )!^ rot. |-assoc = E\(Q fl E\(RQ S)\E)\E

Siseq. E is eq.rel.

C E\{E\{ReS)\EnE\(RQS)\E)\E

(3.118)

=0 so, by (3.120), E\(RQS)\ECT eker{h), contradicting our earlier conclusion that E\(RQS)\E is not in ker(h). Therefore J is a proper relational ideal. In particular, we have E\(RQS)\E £ J and RQS 0 J. By Th. 116, J is contained in a maximal proper relational ideal I. By Th. 115, J is a Peircean ideal of 6 b (E). Let C = {(X,Y) : X,Y C E, X QY e 1} &nd / = (C o E) n (56 (£) x V). By Th. I l l , C e Co6b (£), / is the homomorphism determined by the congruence relation C, I = 0/C, / is the kernel of / , and / maps 6b {E) onto the quotient algebra 6b (E)/C. By (3.115), 6 b (E) /C is isomorphic to a subalgebra of a square relation algebra, so 6b (E) jC G RRA. Let g := h'^f. Then g is functional, for if X, Y C E, (h(X)J(X)} €p (h(Y),f(Y)} e g and /i(X) = h(Y), then (X, Y) G jter(ft) C J C I = ker(jf), so /(X) = f(Y). It is easy to see that E\Q\E £ J, but J C / and Q C S|Q|S, so Q € / . Since 7 is proper, we conclude that £?~Q £ 7. But E~Q = E~(E~E\(RQ S)\E) = E\(RGS)\E, so £ | ( / i e S ) | S $ I and R Q S $ I. Since 7 = ker/, this implies /(7Z) / / ( 5 ) , which, together with h(R) = b and h(S) = c, gives us g(b) = g(h(R)) = f(R) / f(S) = g(h(S)) = g(c). Thus g separates b and c, i.e., 9(b) # S (c).

166

3. GENERAL ALGEBRA

Finally, we note that g is a homomorphism. Suppose x,y € B and choose X,Y C. E such that h(X) = x and h(Y) = y. Then, since / is the homomorphism that maps &b (E) onto the quotient algebra &b (E) /C, we have / ( X U F ) = f(X) U/C f(Y) (where U/C is the operation in the quotient algebra corresponding to U), so g{x + V)= 9(KX) + h(Y)) = g(h(X U Y)) = f(X U Y) = f(X) U/C f(Y), and similarly for the other operations. We have seen that g is a homomorphism mapping © into &b (E) jC € RRA which separates b and c. Since this can be done for anj two distinct elements of 2$, it follows by Th. 88 that 2$ is isomorphic to an algebra in PRRA, but PRRA = RRA, so 93 G RRA. Theorem 120 (Tarski [232]). HRRA = RRA. PROOF. Let SI G RRA and assume there is a homomorphism from St onto an algebra 95 of relational type. Since SI is representable, there is an equivalence relation E € V such that 21 is isomorphic to a subalgebra of &b (E), i.e., 21 6 IS{©6 (£)}, so

03 € H{St} C HIS{66 (£)} = HS{6b (E)} = SH{66 (E)} C SRRA = RRA

(3.40) (3.99) Th. 119 Th. 103

Theorem 121 (Tarski [232]). RRA is a variety; HSPRRA = RRA.

CHAPTER 4

Logic with equality 1. Syntax 1.1. Symbols. Our first task is to define what we mean by a language C. Following Tarski's example in Tarski [226, 227], Tarski-Givant [240, p. 23], we consider extensions of first-order languages having certain features (predicate operators and a predicate equality symbol) that are second-order. Any particular language C is determined by several sets. There are a countably infinite set V = Ra (v) = {vo, vi, V2,..., v,i_i,... }, whose elements are called variables, (the set of variables is the range of a function v : UJ —> V), an equality symbol = (also written 1') whose intended interpretation is equality between individuals, propositional connectives -i and => , the universal quantifier V, a (possibly empty) set C, whose elements are called constants , a (possibly empty) set J-, whose elements are called function symbols, a (possibly empty) set 1Z, whose elements are called relation symbols, a function rank : T U 72. U {=} —> OJ ~{0} that assigns a positive integer (a rank) to each relation symbol, a positive integer to each function symbol, and the positive integer 2 to the equality symbol, that is, rank(=) = 2, o a predicate equality symbol equality symbol = whose intended interpretation is equality between binary relations, o predicate operation symbols +, ~, ;, and ". The items marked with are standard for any language C, while the ones marked with o are the additional features of Tarski's extended languages, obtained by adding operators on binary predicates and a new type of atomic formula that expresses equality between binary predicates. A nice language C is one in which all the sets and elements above, and several others sets constructed below (terms, formulas, etc.), are appropriately distinct and disjoint. This means, among other things, that the following sets are pairwise disjoint: V, {=}, {-.}, { ^ } , {V}, C, T, R, rank, {=}, {+}, {"}, {;}, and {"}. It also means that these sets appropriately contain or are entirely disjoint from the sets of terms and formulas of £ that are defined later. The variables, equality symbols, connectives, universal quantifier, and predicate operators are

168

4. LOGIC WITH EQUALITY

regarded as fixed and consequently the same for all languages. In particular, we chose UJ as the set of variables to be used for equational theories, so it would be natural to include the stipulation that v = UJ1 as part of the definition of nice language. With respect to the fixed set V U {=,->, => ,V, = , +,~, ; , " } , any particular choice of sets C, J-, and 72, and rank function rank will either produce a nice language or it will not. We will usually consider only nice languages. For any given nice language £, let Sym(£) be the set of nonlogical symbols of £, that is, (4.1)

Sym(£) :=CUfU1l.

We regard the language £ as determined by its nonlogical symbols and its rank function. We will express this dependence by writing £ = £(C, J-, 72., rank), leaving the fixed symbols unmentioned. By the way, the rank of a constant in C, if it were defined, would be 0. Assume that £ = C(C,r, 72., rank) is a nice language. Our next task is to construct sets of terms and formulas. 1.2. Terms. The set Tm(£) of terms is the smallest set that contains VUC and is closed under the following rule for forming terms: if / £ T is a function symbol of rank r = rank(/) £ w ~ 1 and ti,... ,tr are terms, then the ordered pair (/, ( t i , . . . , tr)) is a term, usually written fti tr, or / ( i i , . . . , tr), or f(ti -tr), or even ii/<2 in case r = 2. For a nice language, Tm(£) is the universe of an algebra that is absolutely freely generated by the variables V, in which the function symbols T act as operations (although they are actually used only as markers in the formal definitions of these operations). Here is a more precise formulation of this idea for a special case. Theorem 122. Let £ = £(C, J-, 72., rank) be a nice language. Suppose J- = {b,u}, rank(fe) = 2, and rank(ii) = 1. Define operations /3 : (Tm(£)) 2 —> Tm(£) andv : Tm(£) -> Tm(£) by /3(t,t') = (b,(t,t')) and v(t) = (b,(t)) for allt,t' € Tm(£). Then (Tm(£),/?,«) is an algebra of Boolean type that is absolutely freely generated byVUC. If C = {i} then (Tm(£),/3, v, L) is an algebra of group type that is absolutely freely generated by V'. The algebra (Tm(£),/3,u,t) is the t e r m algebra of £. In this example the structure of £ was chosen so that the term algebra is a free algebra of group type on countably many generators. The term algebra for an arbitrary language could have infinitely many operations of arbitrarily large finite rank. 1.3. Predicates. The set II(£) of predicates of £ is the smallest set that contains the equality symbol 1', contains every relation symbol R 6 72. that has rank 2, and is closed under the following rule for forming new predicates from old. If A, B are predicates, then the ordered pairs (+, (A, B)}, {~, {A}}, (;, {A, B)), and (", (A)) are also predicates, usually written A + B, A, A;B, and A, respectively. We usually also write II instead of II(£) when £ is clear from context.

1. SYNTAX

169

Theorem 123. For every language C = C(C, J-, 1Z, rank), II has the properties (4.2)-(4.4). For a nice language, II also has the property (4.5). (4.2)

1' <E n ,

(4.3)

{R : R G 1Z, rank(iZ) = 2} C n ,

(4.4)

if A,B €TI then A + B, A, A;B, A €11,

(4.5)

^3 = (II, +,~, ;, ", 1') is an absolutely free algebra of relational type that is absolutely freely generated by {R : R € 7Z, rank(i?) = 2}.

The algebra ?P = (II, +,~, ;, ", 1') is the predicate algebra of C. Since it is an algebra of relational type, we introduce the standard additional algebraic notation from (3.3) and (3.4), letting 1 = 1' + F , 0 = 1' + F , and, for any A,B G II, A B = A + B and A]B = ~A-^B. Although 1' is the same as = , we will use whichever notation is most appropriate for the context. Note that |II| = u even when 1Z = 0, since II always contains the predicates that can be built up from the equality symbol 1' using the predicate operators. Although the rank function rank is not actually defined on every element of II, it does produce the value 2 when applied to any R e II n H, so we extend the concept of rank to all the predicates in II by asserting that all predicates have rank 2, and may even occasionally express this by writing rank(iZ) = 2 whenever R € II. Even for nice languages, 1Z and II may not be disjoint, since both sets contain the relation symbols of rank 2. However, if there are no relation symbols of rank 2 in 1Z, then, for nice languages, 1Z and II are disjoint. 1.4. Atomic formulas. If R € 1Z U {=}, R has rank r = rank(iZ) > 0, and i i , . . . ,tr € Tm(£), then the ordered pair {R, (ti,.. . ,tr)) is called an atomic f o r m u l a . I t i su s u a l l y w r i t t e n R t \ - t r , o rR ( t \ , . . . , t r ) , o rR ( t \ -tr), o r t\Rt2 in case r = 2. Let AtFm(£) be the set of atomic formulas. AtFm(£) :={R(t1,...,tr)

:

R£1ZU{=}, r = rank(iZ), ti,...,tr

€ Tm(£)}.

Let AtFm + (£) be the larger set, called the extended atomic formulas, obtained by augmenting AtFm(£) with formulas obtained by treating the elements of II as binary relation symbols, together with a new kind of variable-free atomic formula, namely, equations between predicates. If A, B e II, then ( = , {A, B)) is an atomic formula, usually written A = B, called a predicate equation . Predicate equations will be used as the sentences of the formalism Cx (see Table 1), so we define Sent x (£) by (4.6)

Sentx (£) := {A = B : A, B € II},

and define the set of extended atomic formulas thus: AtFm + (£) := AtFm(£) U {t1At2 : A G II,
170

4, LOGIC WITH EQUALITY

Note that two of the sets in this definition overlap. For a nice language C, the intersection of AtFm(£) and {tiAti : A 6 n , i i , t 2 € Tm(£)} is

{tiAti : A 6 K U { = } , rank(A) = 2, tuh £ Tm(£)}, Here are some other obvious consequences of these definitions for a nice language.

AtFm(£) C AtFm+(£), Sent x (£)CAtFm+(£), AtFm (£)n Sent* (£) = 0. 1.5. Formulas. For every nice language C, the set Fm(£) of formulas is defined as the smallest set that contains AtFm (£) and is closed under the following rules of construction. If tp,iji are formulas and x € V, then the ordered pairs {=>, (ip,ip)), (-i, ip), and (V, (x, ip)) are formulas, usually written as ip => ip, -up, and V ^ , respectively. Fm(£) : = f]{X : AtFm(£) C X, if
The set Fm (£) of extended formulas is defined as the smallest set that contains AtFm+(,C) and is closed under the same rules used to construct Fm(£). Fm+(£) : = f\{X : AtFm+(£) C l , if € Fm + (£), and any i £ V , (4.7)

F = ->(vo=vi => v o =vi),

(4.8)

T

(4.9)

ip V i)

= -up => 1JJ,

(4.10)

(pAi/i

= ->(tp =

(4.11)


(4.12)

= V 0 =Vl ^

Vo=Vl,

-,tjl),

*p 3aip

Note that what we have actually defined are operations on formulas. 1.7. Tautologies. Define a binary operation —) and a unary operation — on {T, F} as follows. (4.13)

T^T = T

F^T = T

-T = F

Then {{T, F}, , —) is an algebra of Boolean type. Let St ("statements") be the set of atomic and universally quantified formulas of £ + , that is, St := AtFm+(£) U {Vxtp : ip G Fm + (£), z £ V}.

1. SYNTAX

A valuation is a function mapping St into {T, F}, so v is a valuation iff v : St —> {T,F}. Theorem 124. Let £ = C(C,J-,TZ, rank) be a nice language. (i) Fm + (£) is the closure of St under => and ->, i.e., (4.14)

Fm+(£) = P){X : St C X, if ip,tp £ X then ~^

ip € X}.

(ii) ^Fm + (£), ) «s an algebra of Boolean type that is absolutely freely generated by St. PROOF. Prove (i) by induction. Let S be the set appearing in the right-hand side of (4.14). Now Fm + (£) contains St and is closed under => and -i, but £ is the intersection of all such sets, so £ is included in all such sets, including Fm + (£). This shows S C Fm + (£). For the opposite inclusion, note first that AtFm + (£) C St C £, next, that S is closed under => and -i, and, finally, if ip € £ and x €V, then \/xip e St C £. Since Fm + (£) is the intersection of all sets that contain AtFm + (£) and are closed under => , -i, and Va, for all x 6 V, and S is such a set, it follows that Fm + (£) C S. Since Fm + (£) is closed under =^, we will use "=>" to denote not only the connective => but also the binary operation /3 on Fm + (£) defined by /3(ip,i/>) = (

ip) = , ( , -i) is an algebra of Boolean type because Fm + (£) is closed under -i and => . Prom part (i) we know that S generates (Fm + (£), =>,-i). Furthermore, from the form of the definitions of formulas in Fm + (£) and the assumption that £ is nice it follows that ip => ip f. St and ->ip ^ St for all {T, F} there is a unique function (4.15)

v : Fm+(£) -» {T, F}, +

such that, for all ip, ip £ Fm (£), (4.16)

v(tp) = v(
(4.17)

v(

iP) = v( v(iP),

(4.18)

v(~"P) = ~v(
PROOF. Since a valuation v is a function from the set St of generators of (Fm+(£), =>,-.) to the algebra ({T, F}, ->, - ) , and (Fm+(£), => , ) is absolutely freely generated by St, the valuation v has a unique extension to a homomorphism v from (Fm+(£), =j-, -.) to ({T, F}, ->, - ) , which is what conditions (4.16)-(4.18) assert. D On the basis of the previous Th. 125, we say that a formula (p 6 Fm + (£) is a tautology iff v(tp) = T for every valuation v : St —> {T, F}. The next two theorems provide the basis for the truth-table method of determining whether a formula is a tautology. There is a function st such that st(
172

4. LOGIC WITH EQUALITY

elements of St occurring in ip. The values of a valuation on this set completely determine the value of the valuation on ip. Theorem 126. Let £ be a nice language. There exists a unique map st : Fm+(£) —> Sb (St) such that (i) st(v?) = M i/peSt, (ii) st(ip => iP)=st(tp)l>st(tP), (iii) st(-ip) =st(ip). PROOF. We imitate the proof of Th. 125. Consider the algebra (Sb(St),(3,v) of Boolean type where j3(X,Y) = X U Y and v(X) = X for all X,Y C St. Let / = ({ , -i) into (Sb(St),P,v). U

Theorem 127. Let C be a nice language and

1.8. Free variables. Next we define the set of variables that occur in a term t € Tm(£) or in a formula ip € Fm + (£). The definition is recursive, and there is theorem, left unstated and unproved, that there is a unique function satisfying each of the rules stated below, which could be proved by an argument illustrated by the proof of Th. 125. For each nice language there is a function var(-) : Tm(£) U Fm+(£) -> 56 (V),

uniquely determined by the following rules, in which x £ V, c £ C, f G J-, rank(/) = r, tu...,tr G Tm(£), R G U U II, rank(fl) = r, A,B G II, and V?,V e Fm+(£). var(x) = {x}, var(c) = 0, var(/ii var(Rti

T) T)

= var(ii) U

U var(tT),

= var(ii) U

U var(ir),

var(A = B) = 0, v a r ( y => ip) = var(y) U v a r ( ^ ) ,

var(-up) = var(ip), var(\txip) = var(ip) U {x}.

1. SYNTAX

173

The set of variables that occur freely in a formula

Sb (V), uniquely determined by the following rules. The only difference between the functions var(-) and free(-) is that the universal quantifier adds a variable, but subtracts a free variable. free(p) = var(^)

if

ip £ AtFm+(£),

free(^? => ip) = free(ip) Ufree(^), )) = iree(ip), ) = free(y) ~{x}.

For every formula

) is the set of variables of
Sent(£) C Sent+(£),

(4.20)

Sent x (£)CSent+(£),

(4.21)

Sent x (£)nSent(£) = 0.

For every n < w, let (4.22)

Vn := {v; : i < n}.

In particular, Vo = 0 and Vu = V. We say that a term t £ Tm(£) is an n-term if var(i) C Vn- We define the set of n-terms by (4.23)

Tm n (£) := {t : t € Tm(£), var(i) C Vn}.

Similarly, a formula ip E Fm + (£) is an n-formula if var(^j) C Vn, and that ip is an n-sentence if ip is an n-formula and free(^j) = 0. Whenever m < n < to, let Fm n (£) := {ip : ip E Fm(£), var(v?) C Vn}, Fm n ( m ) (£) := {ip : ip E Fm n (£), free(v?) C V m } , Fm+(£) := {ip :

) C V n }, Fm+ (m) (£) := { v =

Then Fm^(£) is the set of n-formulas and Sent^(£) is the set of n-sentences. Here are a few more simple consequences of these definitions for a nice language £. Fm n ( n )(£) = Fm n (£),

174

4, LOGIC WITH EQUALITY

Fm+n)(£) = Fm+(£), Fm+(£) = Fm+(£), Fm w(0) (£) = Sentw(£) = Sent(£), Fm+ (0) (£) = Sent+(£) = Sent + (£). 1.9. Closures of formulas. Let £ be a nice language. For every tp £ Fm + (£), [(p] is a sentence, called the closure of (p, obtained by universally quantifying

r

if free(^?) = 0, if free(i^) = { v ^ , . . . , vth } and »i <

< i&.

1.10. Substitution. We will define two kinds of substitution, the replacement of a variable or constant by a term, and the interchange of two variables. If £ is a nice language, u € V U C, and t g Tm(£), then the function Sub?*(-) replaces u with t and is uniquely determined by the following rules, in which rank(.R) = r > 0, x £ V, c G C, t £ Tm(£), / £ T, rank(/) = r > 0 , R£TIUU, A,B g II, ti,... ,tr € Tm(£), and (p,ifi € Fm + (£): (4.24)

Sub^(ti) = *,

(4.25)

Sub?(s) = x if « / x,

(4.26) (4.27) (4.28) (4.29) (4.30) (4.31)

Sub^c) = e if « / e, Sub?(/ti...t,) = /(Sub?(ti)...Sub?(<

subr(^ = B) = t SubfiRh.-.tr) = fl(Sub?(ti)...Sub?O = Subr(^) =* Sub?^) = -.Sub?(v),

(4.32)

Sub^(Vti^)

(4.33)

Sub?{V.v) = yxSubt(tp) whenever

The following simple consequences of the relevant definitions can be easily proved by induction. Substitution of a term for a variable has no effect on terms and formulas in which that variable does not occur free. If a variable x does not occur

1. SYNTAX

175

in a term t, then substitution of t for x will eliminate all free occurrences of x in a formula. T h e o r e m 129. Let C be a nice language, x £ V, t,t'

£ Tm(£), and tp £

Fm+(£). (i) //a: g var(t') iften Subf (*') = t'. (ii) //a; 0 free(y>) iften Subf ( £ Fm (£), £ £ Tm(£), and x £V. We say that £ is free for x in

and ->, — is closed under Vj, whenever y £ var(t). The statement "t is free for x in t/j" is short for "i is free to be substituted for x in ip", that is, the computation of Subfi'ip) requires the rule (4.33) only when x £ var(t) or x £ free(tp). Substitution of a term for a variable when that term is not free to be substituted for that variable can have undesirable consequences. For example, a formula making an assertion about x may not make the same assertion about y if y is substituted for x when y is not free for x. For example, the variable y is not free to be substituted for x in MyRxy. The result of making such a substitution anyway is the formula Suby(VyRxy) = VyRyy. The formula VyRxy asserts that x is R to everything, but Sub*(VyRxy) does not say y is R to everything, but instead that something is R to itself. For a more commonplace mathematical example, suppose we define a function / : w —> ui by setting 4

(4.34)

/(«)== X)(i + n)

for every n £ UJ. We may use this formula to calculate values for /. For example, 4

(4.35)

/ (5) = J^ii + 5) = (1 + 5) + (2 + 5) + (3 + 5) + (4 + 5) = 30.

We cannot correctly evaluate f(i) by straightforward substitution in (4.34), because replacing "n" with "i" produces 4

(4.36)

f(i) := ^(i

+ i) = (1 + 1) + (2 + 2) + (3 + 3) + (4 + 4) = 20,

i=i

which, in particular, implies /(5) = 20, contradicting the calculation in (4.35). For every nice language C, every ip 6 Fm + (£), and all i,j € w let Sijtp be the result of interchanging VJ and VJ, that is, simultaneously replacing every occurrence of Vj with Vj and every occurrence of Vj with Vj. For example, if R € H and R is ternary, then Soi(Vvl3vo-RvoviV2) = Vvo3vl-RvivoV2. Suppose i, j , k £ ui, c £ C, 0 < r £ w, R e H U II, the rank of R is r, / G J-', the rank of / is r, and ti,... ,tr G Tm(£). The rules governing the operation Sij are:

4. LOGIC WITH EQUALITY

Sy(Vfc)=Vfc

lfkytij,

Sy (c) = c, ti---* r ) = /(Sy(*i)...Sy(* r )),

fo> => tf) = Syfo>) => Sy (), Sy

) = --Sy (ip),

Sy (Vvj, v) = Vvj, Sy v?

iik^ij.

Notice that one form of substitution uses variables (as well as terms) as inputs, while the other uses indices of variables rather than the variables themselves; for the ways these notions are most frequently used later, this tends to produce notationally simpler expressions. The notion of interchanging (or transposing) two variables was introduced in Maddux [139, p. 188]. Another type of substitution was introduced by Monk [182] and a variation on it was used by Tarski-Givant [240]. All of these variant substitutions are designed to avoid various difficulties that arise when the usual notion of substitution is applied to languages with finitely many variables. For a discussion of these difficulties, see Tarski-Givant [240, pp. 66-71]. These variant notions of substitution are examples of good substitutions, which we now define. Classical substitution may produce undesirable results, and is therefore restricted in various axiom schemata to instances which produce only desirable results. Good substitutions may always be applied, since they simply "respell bound variables as they go along", and therefore always avoid undesirable results. Let £ be a nice language and n < to. A good substitution for n-variable logic is an operation Sub^(-) that associates a formula Sub'9'(<^) 6 Fm^(jC) with every formula

T m n ( £ ) , and which satisfies the following conditions whenever x,y E Vn, c E C, R € TZ, f E T, rank(/) = rank(i?) = r, h,... ,tr E T m n ( £ ) , A,B E U, and
Sub [9l (x) = g(x),

(4.38)

Sub[9l(c)=c,

(4.39)

Sub [ 9 l (/ii

(4.40)

[9l

Sub (i?(ii

(4.41)

Sub[9]{A = B) = A = B,

(4.42)

Sub [9l (-.v?) = -.Sub [9l (<^),

(4.43)

Sub[9](

ip) = Sub^itp)

(4.44)

Sub [ 9 l (V^) = Vh(x)Sub[h](
r)

= )

=> SubM(V>),

2. SEMANTICS

177

(which depends on x,
h(x) E Vn, h{y) = g(y) for every y eVn ~{s}, h(x) i \J{var(h(y)) : y G Vn,h(y)

+ x}.

By a good substitution we mean a good substitution for w-variable logic. For a good substitution, h in (4.44) may depend on all of x, 2 and i? M C M if r = l, (ii) ( = ) ^ = M 1 , (iii) each function symbol f E J- ot rank r = rank(/) is assigned by M to a function fM such that fM : Mr -) M if r > 2 and fM : M -) M if r = l, (iv) each constant c £ C is assigned by M to an element cM £ M. Ii M satisfies all of these conditions except possibly (ii), we say that M is a loose interpretation. Some basic results about interpretations do not depend on the condition (ii) and are therefore stated for loose interpretations. It should be emphasized that every interpretation is a loose interpretation, but not conversely. (This terminological situation arises elsewhere, e.g., every relation algebra a semiassociative relation algebra, but not conversely.) 2.2. Sequences. Recall from (2.393) that "M is the set of functions from u) into M. "M

: = {3 :

s : ui ->

M}.

W

Functions in M are called ^ - s e q u e n c e s of elements of M. For every a>sequence 8 £ "M and every k £ u, we say that Sk is the (k + l ) s t element of s, and express this by writing s = {so, si, 82, S3,...). If s G ^M, k G w, and d G M, then we let s(k/d) be the sequence that coincides with s except that the (k + l ) s t element of s(k/d)

i s d.

T h u s s(k/d)

= (so,si,S2,

S3,...

s(k d)i = < yd

,Sk-i,d,Sk+i,

if k = 1 e

Here is an alternate definition. s(k/d) := (s n {it} x M) U {{k, d)}.

and

178

4, LOGIC WITH EQUALITY

2.3. Denotations. Every loose interpretation M, for a nice language C supplies a denotation for the nonlogical symbols of C (see (4.1)) and for the equality symbol = . A sequence a € w M acts as a further assignment of variables V to elements of M. According to a sequence s, the variable Vj denotes Sj. The set of terms, together with the operations on them determined by the function symbols, forms an absolutely free algebra X which is absolutely freely generated by V U C; see Th. 122. Together, the loose interpretation M, and the sequence a € " M provide a map from V U C into the elements of M. Note that M is the universe of an algebra 31 that is similar to 1, in the sense that 21 has operations that are also indexed by the function symbols. In this case the operation of SI corresponding to / € J- is fM, while the operation corresponding to / in the free algebra X is the one defined in a specific set-theoretical manner which insures that the algebra X is absolutely freely generated by the constants and variables of C Thus the loose interpretation M. and the sequence a together determine a map from the generators of X into the universe of the algebra 21. Since X is absolutely freely generated by V U C, there is a unique extension of this map to a homomorphism from X into 21. This map will be denoted by aM, so the image of the term t under this map is aM(t). (If we include the constants of £ as distinguished elements of X and their interpretations by M. as distinguished elements of SI, then X would be absolutely freely generated by V rather than VUC.) Similarly, if M is an interpretation (and not just a loose interpretation) then there is a unique homomorphism from the predicate algebra *$ = {II, +,~,;, ",1'), which is absolutely freely generated by the relation symbols of rank 2, into the square relation algebra 9te (M). Note that = (also written V) is sent to the correct relation for this to be true, namely, to the identity relation on M. If, on the other hand, M. is a loose interpretation but not an interpretation, then = may not be mapped to the identity relation on M, The images of the binary relation symbols are binary relations on M, as they should be. The non-binary relation symbols are mapped by M. to non-binary relations, but none of those relation symbols is used in the creation of the predicate algebra 5p. For every A € II, let AM be the image of the predicate A under the unique extension of the map {RM :ReH,rank(R)

= 2)

to a homomorphism on the predicate algebra ?p. A more formal expression of these definitions is that for every loose interpretation M for C, every k £ w, every sequence « £ U M , every constant e £C, every function symbol / E T where rank(/) = r, all terms t i , . . . ,tT £ Tm(£), and all predicates A, B g II, (4.45) (4.46)

aM{wk)

:= ak,

M

s (c):=cM,

(4.47)

aM(ft1 -..*,) := / M ( * M ( i i )

(4.48)

(A + B)M :=

(4.49)

(A)M

(4.50)

AMUBM,

:=M2~AM,

(A;B)M:=AM\BM,

sM(U)),

2. SEMANTICS

(4.51)

(A)M :=

179

{AMy\

The last four conditions above assert that the function (-)M is a homomorphism on the predicate algebra *p. In case M. is an interpretation, then we also have (=)M = VM = M 1 , so the function (-)M is a homomorphism from the predicate algebra qj into 9*e (M):

{-)M eHom{¥,mt{M)).

(4.52)

2.4. Satisfaction. Satisfaction is a ternary relation that may or may not hold between a loose interpretation M for a nice language £ = C(C, J-, 72., rank), a formula ip 6 Fm + (£), and a sequence s 6 " M . When it holds, we write M \=

0, terms t i , . . . , tr £ Tm(£), formulas
M \= A = B [s] iff AM = BM,

(4.54)

M\=Rh---tr[s]

(4.55)

M|=(p^i(i[s]iff either M fi tp[s] OT M \= ip[s],

(4.56)

7W^-^[s]

(4.57)

.M |= Vv,^ [s] iffM\=

iff

iff

(^sM(t1),...,sM(tr))

€RM,

Mfi
We say that a loose interpretation .M with domain M is a model of ip 6 Fm + (£), and write .M |= y, iff M \=

C Fm + (£) and any loose interpretation M, we say that M is a model of $, and write M \= $, iff .M |= y; for every ip E $. We say that a formula 7J 6 Fm + (£) is logically valid iff 0 |= ip, that is, iff M \=

ip, then M\= ip. (iii) M\=
180

4. LOGIC WITH EQUALITY

(iv) If M\= A = B then M\=A + C = B + C,M\=C + A = C + B, M \= A = B, M \= A;C = B;C, M \= C;A = C;B, and M \= A = B. (v) IfM\=A = BandM\=A = C then M\= B = C. PROOF. For part (i), suppose

{T, F} by fT if M\=
Then v has a unique extension to v. One may now use the tables in (4.13), the honiomorphism conditions (4.17) and (4.18), and the clauses (4.56) and (4.55) in the definition of satisfaction to show by induction that i)(ip) = T iff M \= ip [s] for every ip E Fm + (£). Since ip is a tautology, we have i)(
(t)=S

(t).

Let

P := {t : t E Tm(£), if s, s E "M agree on var(i) then sM(t) = s'M(t)}. We will show that P contains all variables and constants of £, and is closed under the formation of new terms from terms in P. Consequently, P = Tm(£). Let c 6 C be a constant of £. Assume s,s' E^M. Then sM(c) = cM = M s' {c). Thus c £ P.

2. SEMANTICS

181

Consider an arbitrary variable Vj E V. Assume a, a' G UM and Sh = ajj. whenever vj. g var(vj). But var(vj) = {VJ}, so this assumption simply says that 8j = aj. Then SM(VJ) = % = aj = a'-^v,-). Thus v,- G P . Assume / E T is a function symbol of £ with rank r and t i , . . . , tT £ P. We claim that ft\ tr 6 P . To prove this, assume s,s' 6 WM and sj, = s'j. whenever t r ) = var(ti) U U var(t r ), so it follows that Vjb G var(/ti tr). Now var(/ti * and s' agree on var(ii), a and a' agree on var(ia), . . . , and a and s' agree on var(t r ). From these consequences and the assumption that ti,...,tr 6 P , we may conclude that aM(h) = s'M(h),...,

(4.58)

sM(tr) = s'M(tr).

But then aM(fh

) = fM(sM(h),...,

sM(tr))

= fM{slM{t1),...,slM{tT))

(4.58)

M

= a' (ft!

Thus fti

tr e P .

Now we can show that satisfaction of a formula by a sequence depends only on the free variables of the formula. T h e o r e m 132. Let £ be a nice language and let M be a loose interpretation for C with domain M. For every formula tp E Fm + (£), if sequences s,s' E W M agree onfree(cp), then M\=
iff

M\=tp[a'].

PROOF. Use induction on formulas. Let P := {tp :

{sM(t1),...,sM(tr))£RM, M|=i2ii

For the other atomic case, in which A, B g I I and we wish to show A = B € P , it suffices to observe that, for all sequences a, a' E " M , a and a' agree on freef A = Bj = 0, and the following statements are equivalent by (4.53). M\=A = B[s],

182

4. LOGIC WITH EQUALITY

AM=BM, M \=A = B[s']. If s, s' G "M agree on free((p => ip) = free(ip) U free(^), then the following statements are equivalent by (4.55). M \=tp => ip[s], either A4 \j= ip [s] or M \= ip[s], either M ^ y[s'] or A4 |= V[s'], M

ip,ip € P

\=

t/>[s'].

Assume that y>,^> G P . If s,s' G ^ M agree on free(-xp) = free(y), then the following statements are equivalent by (4.56).

M \= -n/3 Is']-

Assume that s,s' €. ^M agree on free(VVkip). The following statements are equivalent by (4.57). M\=(f

[s(k/d)] for every d G M,

M \=

ip G P, note (a) below

Note (a): For every d € M, s(k/d) and s'(k/d) agree on freeC^vj.^) by hypothesis, but they also agree on {v^} since s(k/d)k = d = s'(k/d)k, so they agree on free(VVs,¥?) U {v^}. But free(y>) C free(VVs,¥:>) U { v *}; s o s(k/d) and s'(k/d) agree D on free (
M \=

[s1].

2.6. Satisfaction and substitution. It is not always true that

This fails, for example, if R € 1Z, rank(i?) = 1, ip = ip = Rvo, M = {a, b}, a ^ b, RM = {b}} se"M, and so = b, since

M ^V

2. SEMANTICS

The trouble is that the universally quantified variable vo occurs free in the formula ip. If this is not the case, then, as we see next, no such counterexample is possible. Theorem 134. If £ is a nice language,

PROOF. "We wish to show that M \= Vvk( rp) => (

Vvhip)[s] for every loose interpretation M and every sequence s GUM, i.e., either M \/= y vji {tp => tj)) [a] or M \fc
M\=VVk{il>)[a]t

(4.60)

M | = tfi[s].

Let de M. Erom (4.59) and (4.57) we get (4.61)

M\=

i/> [a(k/d)].

Prom (4.61) and (4.55) we conclude that (4.62)

either M £

By hypothesis, vj. ^ free(^)), so * and a(k/d) agree on free(y>). It follows by Th. 132 and (4.60) that (4.63)

M\=V[a(k/d)].

By (4.62) and (4.63), we get M \= i/f [a(k/d)]. We proved this for every d 6 M, so it follows by (4.57) that M 1= Vv&^ [«]. The next two theorems are lemmas for the proof of Th. 137. The first one states the connection between the substitution of a term for a variable and the alteration of a sequence. Theorem 135. Suppose £ is a nice language, M is a loose interpretation for £ with domain M, a GWM, t,t' G Tm(£), and keu. Then (4.64)

s(k/8M(t))M(t').

sM (Subr*(O) =

PROOF. Use induction on terms. First consider the case when t' is a variable, say t' = v r o . Then, by the relevant definitions, 3 M (v ro ) = sm

if m / k

j

11 m — K

(t;

and \9m [aM(t)

iim^k ifm = ,

so (4.64) holds when t' is a variable. In case t' is a constant, say t' = c € C, we have

184

4. LOGIC WITH EQUALITY

A s s u m e t h e t h e o r e m holds for t e r m s i i , . . . ,tr s y m b o l of r a n k r. T h e n M

(/ti

tT)) = SM(f

G T m ( £ ) and / 6 f ,

SubVth ( t l )

a function

Sllb^ (ir))

= fM(s(k/sM(t))M(t1),..

.,s(k/sM(t))M(U))

s(k/sM(t))M(ft1---tr).

=

D T h e o r e m 136. Suppose £ is a nice language, M is a loose interpretation for £ with domain M, s G UM, t G Tm(£), ip G Fm+ (£), k G u, and t is free for Vk in ifi. Then

M\=
(4.65)

PROOF. The proof is by induction. Let k G co, t G Tm(£), and P := {tp : i> G Fm+(£) and for all s G UM, M

iff Af|=Subrfc(V)M}-

We will show that P has the closure properties listed in the definition on p. 175 of "is free to be substituted for", which implies that ip G P, as desired. First we show that if vfe g free(V>), then

(4.66)

ipeP.

Let s € " M . Then the following statements are equivalent: M \= tp [s]

note (a) below

fe

M \= Sub^ (ip) [s]

note (b) below

M

Note (a): s(k/s (t)) and s agree everywhere except possibly at k, but vj, is not agree on the free variables of ip. free in ip by assumption, so s and s(k/sM(t)) Apply Th. 132. Note (b): By Th. 129, Sub*" (ip) = ip. This completes the proof of (4.66). Next we show that every atomic formula is in P. Consider an atomic formula A = B with A,B e Tl. Then A = B € P by (4.66). Next consider an atomic £ Tm(£). Let formula Rti T where R € U U II, r = rank(iZ), and ti,...,tr s G uM. Then the following statements are equivalent: M

\=Rti---tr[s(k/sM(t))], ,

8(k/8M(t))M(tr))

G -RM, i?^,

Th. 135

2. SEMANTICS

Thus every atomic formula is in P. Next we will show P is closed under = and universal quantification with respect to variables that are not free in t. Suppose «/>,£ G P. Let s GWM. Then we have M\=1>[s(k/sM(t))]

iff

M\=SUbvt-(1>)[s],

M\=S[8(k/sM(t))] iff A<|=Sub^ (£)[*], so the following statements are equivalent:

M^sub^a, => e)M, M ^ Sub"fe ('4>) [s] or M \= Sub"<" (0 [s], M ^^[s{klsM{t))\

or M\=^[s(k/sM(t))],

V,X G P

Thus «/> = J - ^ 6 P , which completes the proof that P is closed under . For closure under -i, assume ip £ P and let s £ "M. Then the following statements are equivalent:

MY=ip[s(k/sM(t))], Thus -it/; E P, so P is closed under -i. Finally, we assume ip 6 P and y ^ var(i), and show that Vyip G P- Suppose y = VJ. If fc = j then y = v^ and V& ^ free(VJ/-i/1)! s o Vj,^ G P by (4.66). Therefore, assume k ^ j . We wish to show that VVj V1 G P- Let s GWM. Then the following statements are equivalent. X |= ip [s(k/sM(t))(j/d)] M

X |= ip [s(j/d)(k/s (t))] M

for all d G M,

(4.57)

for all d G M,

note (c) below

M\=tp [s(j/d)(k/s(j/d) (t))] for all d G M, M \= Sub^fe (V>) [sO'/d)] for all d G M,

note (d) below note (e) below,

>f ^ V v . Sub^(V>)H, M^Sub^(Vv^)[s],

(4.57) (4.33)

Note (c): s{k/sM(t))(j/d) = s(j/d){k/sM{t)) since k ^ j . Note (d): Since M vj f var(t), we have s {t) = s(J/d)M(t) by Th. 131. Note (e): We can apply the assumption that tp G P, with s(j/d) in place of s. This completes the proof that P is closed under universal quantification with respect to variables that are not free in t. Since has the required closure properties, we conclude that ip G P, so (4.65) holds. Theorem 137. Let C be a nice language, t G Tm(£), k G u>, andip G Fm + (£). If t is free for Vk in tp, then \= Vvj,^ => SubJh (ip).

186

4. LOGIC WITH EQUALITY

PROOF. Let M be a loose interpretation for £ with domain M, and let s £ M. We wish to show that M \= VVfev? =^ Sub^fc(i^) [s]. For this purpose, assume .M |= V v ^ M - : t follows from this by (4.57) that M \= ip[s(k/sM(t))]. We may apply Th. 136 since t is free for vjt in cp by assumption, so M. \= SubJ'*' (<^) [s], as desired. U

The hypothesis that t is free for x in ip is necessary, since there are formulas of the form Vxip => Sub?(ip) that are not logically valid. For example, \i R £ 1Z, and rank(iZ) = 2 then => Siiby(3yRxy) => 3ySuby{Rxy) = Vx3yRxy => 3w.RSub* = Vx3yRxy => 3yRyy. However, \/x3yRxy => 3yRyy is not logically valid, since if M = {a, b}, a ^ b, and RM = {(a,b) ,(b,a)}, so that RM is simply the diversity relation on the two-element set M, then M \= Mx3yRxy (in M, everything is different from something), but M ^= 3yRyy (in M it is not the case that there is something different from itself). 3. Axiomatization and formalisms A ("syntactical" or "uninterpreted") formalism (see Tarski-Givant [240, §1.6]) is a pair (S, h), where S G V is a set (whose elements are called "sentences" or "formulas") and h is a binary relation (called "provability" and read "proves") which relates subsets of E to elements of S, that is, h C Sb (S) x S, and which satisfies the following conditions for all $ , $ C S and all ip,ip £ S. (4.67) (4.68) (4.69) (4.70)

if f £ * then $ h
Consider two formalisms T\ = (Si, hi) and Ti = (S2,h2). We say that T\ is a subformalism of Ti (and Ti extends T\) if Si C S2 and $ hi ip implies <& \~2 tp whenever $ C S i and ip £ Si, that is, hi C I-2. For every one of the formalisms that we will consider, such as the formalisms C, C+, and Cx of Tarski-Givant [240], the set of sentences is a subset of Fm+(£) for some nice language £. Consequently the semantical notions defined for any nice language C are available. Accordingly, and following Tarski-Givant [240], if Si, S2 C Fm + (£) and T\ is a subformalism of T2, we say that the formalisms T\ and J~2 are equipollent in means of expression if for every ip2 £ S2 there is a ipi 6 Si such that ip2 = ipi, and that T\ and Ti are equipollent in means of proof if $ l~2 v? implies $ hi (p whenever f C E i and j j g E i , i.e., h 2 n(5fo(Si) X S I ) = K .

3, AXIOMATIZATION AND FORMALISMS

187

If T\ and .Fa are equipollent in means of proof then, for every $ C Si and every p € Ei, we have # ha