Algebra Universalis, 7 (1977) 191-194
Birkh~iuser Verlag, Basel
3-3 lattice inclusions imply congruence modularity RALPH FREESE* and J. B. NATION*
The complexity of a lattice polynomial is defined inductively, with variables having complexity 0. If p = p l v ' " v O k or p = p l A ' ' ' / X O k is the canonical expression of the polynomial O, then the complexity c(p) = l + m a x { c ( p ~ ) : l < - i - k } . An n - k lattice inclusion is an inclusion of the form p <-o- with c(p)<- n and c(o-)--- k. In this note we use the main result of Day [1] to show that if all the congruence lattices of algebras in a variety satisfy a fixed, nontrivial 3-3 lattice inclusion, then they are all modular. Let q be an element of a distributive lattice D. Then D[q] will denote the lattice obtained by "doubling" the element q, i.e., D[q] = (D -{q}) U {(q, 0), (q, 1)} ordered b y a < b i f a , beD-{q}anda
(*) A V ~j ~- V A ~ I
J~
K
Lk
fails in F L ( X ) if and only if (Vi)(:l])(Vk)(3l) var (rrij)N var (r
= 0
and
(Vk)(al)(Vi)(aj) var (Trii) N var (r
= 0.
Presented by G. Gr/itzer. Received December 15, 1975. Accepted for publication in final form June 17, 1976. * This work was supported in part by NSF Grants Nos. MP8 73-08589 A02 and GP-29129 A-3. 191
192
RALPH FREESE AND S. B. NATION
A L G E BRA UNIV.
Proof. By r e p e a t e d applications of W h i t m a n ' s conditions, we obtain that the failure of (*) is equivalent to (Vi)(3j)(Vk)(31) var (~rii) N var (o'kl) ----~ and
(Vk)(31)(Vi)(3j) var (~'q) N var (o'kt) = O and
(Vi)(3j)(Vx ~ var ( ~rij))(Vk )(31)x~: var (o'kt) and
(V k )(31)(V x ~ var ( O'k~))(V i )(::lj)x ~. var (~'ij). Since the third and fourth conditions are consequences of the first two, the l e m m a follows.
T H E O R E M . Let e be a nontrivial 3-3 lattice inclusion. Then there exists a finite distributive lattice D and an element q ~ D such that the lattice D[q] fails to satisfy e.
Proof. It is easy to see that we m a y assume that e is of the form (*). Let X be the set of variables involved in e. T h e n we may assume that e holds in the free distributive lattice FD(X), for otherwise it fails in 2, which is of the f o r m D[q]. Let h be the evaluation of the left-hand side of e in FD(X), and let p be the, evaluation of the right-hand side. Let 0 be the smallest c o n g r u e n c e on FD(X) which identifies h and p. W e let D = FD(X)/O and we Iet q be the c o n g r u e n c e class containing A, i.e., q = A/0 ( = O/0). In order to show that e fails in D[q], we interpret the variables as follows: for x e X , let ,g=x/O if x/O;~q, and ,~ = (q, 1) if x/O=q. For o'~FL(X), let ~ denote the image of o- in D[q] under this interpretation, and let 6" d e n o t e the evaluation of o- in FD(X). We shall show that for each i6 I, V z ,-rq > ( q , 1) and dually for each k e K, At~ ok~ < (q, 0). T h e desired conclusion will then follow. First observe that in FD(X), O = VK A~.~d'k = A L ~ VKd'k.t(k) w h e r e L • denotes the set of all functions f:K---, [_J Lk such that f ( k ) e Lk. By the l e m m a , for each i ~ I there exist / ~ Ji and fi ~ L K such that var (~0) f'l [ U K vat (~rk, fi(k))] = 0 . It follows that in FD(X), "Trq$ VK d'k,t.(k), arid consequently V~, ~'~i~p, for each i~L
Vol 7, 1977
3-3 Lattice inclusions imply congruencemodularity
193
Recall that in a distributive lattice, if a<--b and a<-czg b, then (a, c ) ~ c o n (a, b). Now in F D ( X ) we have h_
(q, 1). Dually we obtain AL~ crkt < (q, 0) for each k 6 K. This completes the proof. C O R O L L A R Y . Let e be a lattice inclusion. The following are equivalent: (i) e fails in D[q] for some distributive lattice D and some q ~ D. (ii) For some m, n >-2 e implies the inclusion
l<--i~n\
l~j-"-~m
I
m
l<-i~--n
where
2,= V xkv V k~i
l<--j<--m
yj
and
~i =
V l<.i<--n
x, v V yk. k~j
(iii) e implies some nontrivial 3-3 lattice inclusion. Proof. In [1] Day shows that (i) is equivalent to (ii), and that (ii) implies (iii) is obvious. By the Theorem, (iii) implies (i). Not every nontrivial lattice inclusion implies a nontrivial 3-3 inclusion. Consider the conjugate inclusion for the splitting lattice N6 ot~ [2]:
(v) y/~((x/~(wv(xAz)))v(zA(wv(x/~z)))) --< x v ((x v y v (w A ( x v z))) A (z v (w A (x v z)))).
If v implied some nontrivial 3-3 inclusion, then it would imply some inclusion /3 of the form of condition (ii) of the Corollary. By [1] /3 is the conjugate inclusion f o r a splitting lattice of the form D[q] where D is a finite Boolean algebra and q is a doubly reducible element of D. Since D[q] fails /3, it would fail v, and N6 would be in the variety generated by D[q]. By Jdnsson's Lemma we could imbed N 6 in D[q], which however is easily seen to be impossible. Thus v does not imply any nontrivial 3-3 lattice inclusion. Similarly, the conjugate inclusions for the lattices Oo, O~, and Q4 of [2] and their duals do not imply any nontrivial 3-3 inclusions.
194
RALPH FREESE AND J. B. NATION
REFERENCES [I] A. DAY, Splitting lattices and congruence modularity, preprint. [2] R. N. McKENZlE, Equational bases and non-modular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1-43.
University of Hawaii Honolulu, Hawaii U.S.A. Vanderbilt University Nashville, Tennessee U.S.A.
