Algebra Universalis, 8 (1978) 336-348 Birkh~iuser Verlag, Basel

Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties

RALPH MCKENZrE 1

Abstract. We introduce new sufficient conditions for a finite algebra ~ to possess a finite basis of identities. The conditions are that the variety generated by ~ possess essentially only finitely many subdirectly irreducible algebras, and have definable principal congruences. Both conditions are satisfied if this variety is directly representable by a finite set of finite algebras. One task of the paper is to show that virtually no lattice varieties possess definable principal congruences. However, the main purpose of the paper is to apply the new criterion in proving that every para primal variety (congruence permutable variety generated by finitely many para primal algebras) is finitely axiomatiza- ble. The paper also contains a completely new approach to the structure theory of para primal varieties which complements and extends somewhat the recent work of Clark and Krauss.

Introduction

This paper is concerned primarily with the finite basis p rob lem for finite

algebras: Let t ing ~[ be a finite algebra of finite type, is the variety genera ted by

finitely axiomat izable? Or, in o ther words , does there exist a finite basis of identities t rue in ~[ f rom which all identities true in ~ can be logically der ived? I t

is known that all finite groups, rings, and lattices have a finite basis ([10], [6], [7]). The multiplicative semigroup of 2 -by-2 matr ices over a 2 -e lement field has a

6 -e l emen t subsemigroup with no finite basis [11]. The result for lattices was

cons iderably general ized by Baker , who p roved that every finite a lgebra whose

genera ted variety is congruence distributive has a finite basis [2].

A n algebra is called C.P. if its genera ted variety has permut ing congruence

relations. I t may be that every finite C.P. a lgebra has a finite basis. The only real

evidence for this is a lack of cont ra ry evidence. In fact, only a handful of non-

f ini tely-based finite algebras have been discovered. Still it is possible, and would

certainly be a nice compan ion to Baker ' s theorem, having as corollaries the results for groups and rings.

x The preparation of this paper was supported in part by NSF Grant MPS 74~ "The author thanks the referee for an unusually thoughtful critique of the paper, and Stanley Burris for many helpful conversations while the work was in progress."

Presented by G. Gr~itzer. Received April 30, 1976. Accepted for publication in final form June 21, 1977.

336

Vol 8, 1978 Para primal varieties 337

In [8] we proved that every finite C.P. algebra whose identities are maximal (i.e., an equationally complete algebra) has a finite basis. Here we shall prove that a C.P. variety generated by a finite set of such algebras, or indeed by a finite set of para primal algebras, has a finite basis. We get this as a corollary of an easy general result which is worth remembering for future applications: I f a variety has the properties (1) ~ has only finitely many subdirectly irreducible algebras, all finite; and (2) ~ has definable principal congruences; then ~ is finitely based.

B. Jonsson once asked whether (1) alone is enough to ensure a finite basis. Also whether the quasi-variety generated by a finite set ~K of finite algebras, that is SP(~K), must be finitely axiomatizable. Clearly, if the second answer is yes then so is the first, and this in turn would subsume Baker's theorem, but not the analogous results for finite groups and rings. Jonsson's first question is still open. The second question has been answered negatively in [9].

We conclude the paper with some constructions showing that the only lattice varieties having definable principal congruences are the two varieties of distribu- tive lattices.

1. Definit ions and results

In this paper algebras will be denoted ~[ = (A, F~ (i ~/)) . A is a non-void set, I is a finite set, and each F~ is an operation over A of finite rank p(i). The function O is the type of ~. Normally all algebras mentioned in one context are assumed to have the same type. ~ is finite if A is.

DEF INI TI ON 1. (1) A variety OF is residually * finite iff there is a finite integer n such that no subdirectly irreducible algebra in OF has a cardinality greater than n.

(2) ([1]) A variety OF has definable principal congruences iff there is a formula ~b(u, v, x, y) in the first order language of ~ so that for all ~ ~ OF and a, b, c, d A : q~"t(a, b, c, d) iff (a, b) belongs to the principal congruence O(c, d).

(3) A first order formula Oh(u, v, x, y) is called a congruence formula if it is existential positive and the implication ~ y --~ u -~ v holds in all algebras.

L E M M A 2. (1) I f a variety has definable principal congruences then some congruence formula defines principal congruences over the variety.

(2) I f OF is a locally finite variety and a congruence formula r defines the principal congruences in all finitely generated (equivalently, finite) members of OF, then r defines principal congruences in OF.

(3) I f ~ is locally finite and the class of finite algebras in OF is residually * finite, then OF is residually * finite.

338 RALPH MCKENZIE ALGEBRA UNIV

T H E O R E M 3. Every residually * finite variety with definable principal con- gruences is finitely axiomatizable.

Proofs. The lemma is elementary. Concerning Lemma 2(3), see [13]. For the theorem, let or be a variety, residually __< n, and ~b a congruence formula defining principal congruences in or. Let ~ be the first order sentence stating that "for all x, y, 4~(-,-, x, y) is a congruence and 4~(x, y, x, y)." (Here our restriction to varieties of finite type comes in.) Notice that an arbitrary algebra ~ of the type of ~ satisfies 4' just in case ~b defines the principal congruences in ')1. Thus or~ ~0.

Now consider the sentence

/3(n):Vu, v, Xo . . . . . x,,(A {q~(u, v, xi, xj): i <]<-- n}---~ u ~ v).

This holds in or; in fact, if ~[ ~ - /3(n) , then Pl must have a subdirectly irreducible homomorphic image whose cardinality exceeds n.

Let Z be a finite set of identities such that or~ Z, Z ~ ~b,/3(n), and every algebra of cardinality --< n satisfying X belongs to or. (There are essentially only finitely many such algebras, so it takes only a finite set of identities to exclude those not belonging to or.) We claim that Z is a full set of axioms for or. It clearly suffices to show that every subdirectly irreducible model of X has cardinality -< n. So suppose ~ is a S.I. model of ~. Since ~ ~ ~/, and is S.I., there are a, b ~ A such that q~"~(a, b, x, y) is equivalent to x ~ y. Since ~ ~/3(n) as well, we have the result.

We remark that a residually small variety (i.e., residually < A, for some cardinal A) with definable principal congruences must be residually * finite (see [1]). Thus the hypotheses of Theorem 3 can be visibly weakened.

DEFINITION 4 ([4]). A variety or is directly representable by a set At of finite algebras if At c_ or and every finite algebra in or is isomorphic to a product of algebras, each belonging to At.

T H E O R E M 5. Every locally finite variety, directly representable by a finite set of finite algebras, is residually * finite and has definable principal congruences.

Proof. Let or be a locally finite variety directly representable by the finite set At of finite algebras. That or is residually �9 finite follows by Lemma 2.3. To show the other we argue by way of contradiction. By Lemma 2.2, we can assume that for every congruence formula ~b, there is a finite ~ e % a, b, c, d ~ A, such that (a,b)~O(c,d) and ~ c b ( a , b , c , d ) o Note that the set q~ of all congruence

Vol 8, 1978 Para primal varieties

formulas is closed under finite disjunctions, and in any algebra ~ ,

339

(a,b)~O(c,d)r V 4~(a,b,c,d). da~c~

By these facts, there obviously must exist an infinite sequence q~o, q~ . . . . of congruence formulas, and sequence ?~o,~t . . . . of finite algebras in 7~, and an, b,, c,, d, ~ A,, satisfying:

4,. ~ 4,.+,, ~ . ~ 4,~ b., cn, d.)

~n ~ ~,0n(a., b., cn, d.).

We can assume that 2[. = ~ t . ~ i where 23i ~d~ and In is finite. Now there are only finitely many pentuples (a, b, c, d, 25), 2 ~ , a, b, c, d~B.

Hence there are n < m so that

{(a.(i), bn(i), c.(i), dn(i), ~,): i6 I.}

= {(am(j), bin(j), cm(j), din(j), ~i): j ~ I,~}.

But since ~b,. is an existential positive formula and ~ln ~ ~bm(a., b., c., dn), the above equality implies that 2~,. ~ ~b.,(am, b,., cm, d.,), a contradiction.

DEFINITION 6. (1) A variety ~ is C.P. iff all congruences on ~V-algebras permute; equivalently, there is a term M such that the "Mal'cev identities" M(x, x, y ) ~ y, M(y, x, x) ~ y hold in ~F. An algebra is C.P. iff its generated variety is C.P.

(2) An algebra is para primal iff it is finite, C.P., and each of its non-trivial subalgebras is simple. (A "trivial" algebra is a 1-element algebra.)

(3) A variety is para primal iff it is C.P. and generated by a finite set of para primal algebras.

T H E O R E M 7. Every para primal variety is finitely axiomatizable.

2. Structure of para primal varieties

The study of para primal algebras was begun by Clark and Krauss [3] who proved that ~ is para primal if and only if a certain combinatorial property is possessed by all subalgebras of finite powers n~. Clark and Krauss ([3], [4]) developed an elaborate and elegant structure theory for varieties generated by a

340 R A L P H M C K E N Z I E A L G E B R A U N I V

single para primal. Among other things, they proved that such a variety is directly representable by a finite set of finite algebras.

The author in [8] made an observation which, if pursued further, clarifies considerably the results of Clark and Krauss. We shall do that in this section, at least up to the point where we can obtain Theorem 7 as a corollary of Theorems 3 and 5. The results of this section are all new, but 11, 12, 14, 15 and 17 are very close to results of [4].

Throughout this section, ~ denotes a fixed para primal variety; ~ i . . . . . ~ , are (finite) para primal algebras generating ~; and M denotes a term in the language of ~ such that the Mal'cev identities (Def. 6.1) hold in ~. Unless otherwise noted (as just below), all algebras will be of the same type as ~-algebras.

If ~ = (A, +, - ) is an abelian group, a K-ary operation f over A is called linear over s~ if f is a homomorphism "~t ---> ~t; and f is called affine over .d if there is a linear g and a c ~ A such that f ( i ) = g(:~) + c for all :~ ~ "A.

D E F I N I T I O N 8 ([8]). An algebra ~ = ( A , . . . ) is affine if[ there is an abelian group ~ = (A, +, - ) such that all the operations (hence all the polynomials) of are affine over ~ . (We say that ~ is affine over ~.)

We recall from [8] a fact that is rather obvious: If ~ is affine over ~t, and also C.P., then x - y + z is a polynomial operat ion in ~ ; in fact, it is the only polynomial satisfying the Mal'cev identities for congruence perrnutability. The attine C.P. algebras satisfying the Mal'cev identities for M form a finitely axiomatizable variety. It is important that this variety has axioms of a certain type. (Robert Trent is responsible for a final simplification which gives the following axioms.)

D E F I N I T I O N 9. Let g o , - . - , ~ , be the basic operation symbols in the lan- guage of ~. Let x be an integer exceeding 3 .max{p(~i):O<-i<--l}. Let ~,= (Zo . . . . , z,_l), where x, y, Zo . . . . . z~_l are distinct variables. Let So . . . . . sl be the terms M(zo, ~M(f~,~, frO, M(Ofi, Oi~,O~)) in which ~ = ( z l . . . . . z~), 13= (za+l . . . . . z2~), ~ =(z2a+l . . . . . z3~), a =0(~7~), and M(a, ~, if) denotes the a- tuple

(M(zl, za+l, z2~+1) . . . . , M(z,,, z2~, z3~)).

Finally, let to . . . . , t~ be a list of aU the terms t(x, ~.) obtained by substituting x for zj in si, as ] ranges over 1 . . . . . t< -1 and i ranges over 0 , . . . , I.

Vol 8, 1978 Para primal varieties 341

L E M M A 10. (1) A n algebra ~ ~ ~ is affine iff it satisfies the identities t~(x, ~)-~

t~(y, if) (O<-i<-m). (2) I f ~ ~ ~ and a, b ~ A and ] <- m, there is ~ ~ " A such that t'~ ( a, ~) = t~( b, ~).

Proof. Let N ~ ~ satisfy the prescribed identities. As a consequence of these, each te rm si(2) is in ~ independent of z~ . . . . . z~_x. Thus we have (in N),

M(zo , r ~, frO, M(~,fi, ~?,~, ~ ,~))

= M(zo, r a, a), M(~,a, ~,a, r

= M(zo , r r

ZO.

Taking Zo = ~?~M(t~, ~, ~) gives:

r ~, ff~) ~ M(CT~t~, r r O<_i<_l. (A)

This means that M ~'~ is a homomorphism 3~[ ___> ~. Since M is a term, the following identity must likewise hold in ~ :

M ( M ( u o , Vo, Wo), M ( u l , I")1' W1), M(u2, v2, w2))

-~ M ( M ( u o , ul, u2), M(vo, vl, v2), M(wo, wt, w2)). (AM)

Now choose any a ~ A and put x + y = M(x , a, y), - x = M(a , x, a). From (AM) and the Mal 'cev identities it is straightforward to show that ~d = (A, + , - ) is an abelian group such that M(x , y, z) = x - y + z. Then the identities (A) imply that is affine over s~.

We leave to the reader to verify that our identities do hold if ~ ~ ~ is affine over some d (recalling that M(x, y, z) = x - y + z relates ~ with ~ in this case).

As for (2), let ~ ~ 9 , a, b ~ A , and j -< m. We can assume that x actually occurs in t r Harking back to Def. 9 we have to consider three cases depending on whether x replaced one of the variables in fi, or in ~, or in ~, in obtaining tj f rom si =si(Zo, a, 6, ~). If in fi or k, it is sufficient to note that 9 ~ s~(zo, fi, ~, ~ ) ~ s~(Zo, a ,a , f f , )~Zo . If in 13, let us suppose for simplicity that t j ( x , ~ ) - s~(Zo; ~; x, z~+2 . . . . , z2,; ~) . Now take c~, = a for all u < K except u = 1; put ca = b. Calculate that tj(a, ~) = tj(b, ~) = a.

Now we can derive a crucial theorem that helps to explain why para primal varieties have a very nice structure theory.

3 4 2 R A L P H M C K E N Z I E A L G E B R A U N I V

T H E O R E M 11. (1) Every finite ~ is isomorphic to a direct product ~ * x ~ x . . . x ~ (r--0) where ~* is affine and ~2~ x . . . . . ~r are non-affine sub- algebras of ~ , . . . , ~ , . (I.e., each ~ is included in some ~i.)

(2) Every congruence on ~ ' = ~ * x . . . x ~ (as above) is equal to a product 0*x 0~ x . . - x Or (O* on ~*, O~ on ~ ; thus each O~ is the identity or the universal relation on ~ - since ~ is simple).

(3) The system ~*, ~1 . . . . . ~ is unique up to isomorphism and permutation of ~ . . . . . ~,. In fact, the representation (1) is internally unique in ~. (There is only one system of factor relations giving such a representation.)

Proof. We begin with (2). Let 0 be a congruence on ~' . What we use is that ~* is afIine, ~[~ . . . . . ~ are non-affine and simple (and all are in ~). Let p*, Pl . . . . . Pr be the projections on ~['; 7r*,Trt . . . . . ~rr their kernels; and put 4r,= �9 r * n N {7r~: l<-v<-r, v ~ u } for l<-u<_r.

Claim 1: For each u, either 0_cTr~ or 4r ,~0. To prove it, suppose 0gTr~. Then 0 1 ~'~ = 2(A') by simplicity of ~ and congruence permutability of ~ ' . By 10.1, pick tl and a, b e A~, ~ "A~, such that ?~, ~ ti(a, ~) # t~(b, ~). Since 0 [ ~r~ = 2(A'), we can choose a, b e A ' so that p~(a) = a, p,(b) = b and a0b. By 10.2 we can choose ~:e "(A') so that p~(~)= ~ and, setting e = t~'(a, ~:), f = t~'r(b, ~.), we have (e, f) ~ ~'~. Clearly, (e, f) ~ (0 n 7?~)\zr,. Thus (0 n ~-~) [ r = 2(A'). However, clearly [(0 n ~ ) [ ~r~] n ~r~ = 0 n ~'~, because ~'~ n ~-~ is the identity relation. Thus 0 _~ 4r~ as claimed.

To finish the proof of (2), it is enough to show Claim 2: 0 = (0 [ ~-*)n n ~ " ( o [ ~~). So let I = { u : 1<_ u<_r, 0 ~ ~',} and 7ri= n {~'.: u ~/}. By Claim 1, 0 ~ V {~'o : v~ /} = 7r* n 7r~, and of course 0 _c 7r~. So

1 , r

0 = o I (#*n #,)=(o I #*)n #,= o l # *n n (o I #.). u

Now for (1), let ~ be finite. By the usual abstract nonsense, ~ HSPf{~I . . . . . ~3,}. That is, there is a finite product, each factor one of the generating algebras, that has a subalgebra that maps homomorphically onto ~. It is known, and easy to prove, that in a congruence permutable variety, a subdirect product of a finite system of simple algebras is isomorphic to a product of some of the factors. Since all subalgebras of {~1 . . . . ,~3n} are simple, it follows that

~ HPfS{~31,... ,~ ,} . So we can write

~---(~:o x ~ :~x . . . x~,)lO

Vol 8, 1978 Para primal varieties 343

where ~t . . . . . ~, are non-affine algebras in S { 2 3 1 , . . " , 2 3 n } , S ~ 0 , and ~o is a finite product of affine algebras belonging to S{23~ . . . . . 23,}. (See Corollary 12.)

By a simple application of (11.2), already proved, concerning 0 we get the desired representation (1). ~* is a homomorphic image of ~o, and ~ . . . . , ~ are among ~ . . . . . ~,.

As for (11.3), it follows easily from (11.2).

C O R O L L A R Y 12. The affine algebras in ~ are generated as a variety by the aIfine subalgebras of {23~ . . . . . 23,}.

Proof. Since ~ is locally finite, it suffices to remark that if ~ is a finite affine algebra in ~, then in (11.1) r = 0 , and the proof of (11.1) shows that ?[ ~.)I* is a quotient of a finite product of anne subalgebras of the 23's.

DEFINITION 13. Suppose that ~ [ ~ is affine over ~/. Let 0 denote the identity element of d . For each of the basic operations ~i of ~, put ff~(~i)= Oi (a ) -~ i (0 , 0 . . . . . 0). Put ~[T=(A, ~?~ (i<l)). Thus ~l T is linear over sg, and we call it the linear derivate of P[.

L E M M A 14. Let 4[ ~ be affine. Then p[,re ~ and g T is determined up to isomorphism by ~[ (independently of ~) .

Proof. ~T~_ (1Ix 11)/0 where (x, y)0(u, v) iff x - u = y - v iff M(x, u, x) = M(y, v, x).

T H E O R E M 15. Let ~ ~ be subdirectly irreducible and affine. Then 4[ is para primal and there is 23~S{231 . . . . . 23,} such that 23 is affine and ~['~- 23v.

Proof. It suffices to prove it under the assumption ~ is finite, for then, by Lemma 2.3, the affine subvariety of ~ has no infinite subdirectly irreducible algebras. We just outline the proof, as it is only a streamlined version of what appears in [4; Thm. 1.11(iii)], simplified by the knowledge that we are dealing with affine algebras.

Let ~ be S.I., and affine. As in the proof of Theorem 11 we can assume that

=(~o x . . �9 x~s)/e

where go . . . . . ~s ~ S { 2 3 1 , * �9 " , 23n} are non-trivial and affine. Let ~ , be affine over ~/, and write 0 for the zero-element of any abelian group. Notice that congru- ences in affine C.P. algebras are just congruences in their groups such that the

344 R A L P H M C K E N Z I E A L G E B R A UNI 'V

coset of 0 is closed under the linear operations ~ . Since each ~u is simple, s( , has no proper 6V-operator subgroups. Let R =0/~, O~ = 0/r for u -< s. Let I be a minimal subset of { 0 , . . . , s} with the property that R N ~ {O,: u~/3 = (0). Then for all v ~ L Q c_R+~{Qu: u~/3. (Since Oo is ~T-simple.)

In combinatorial terms, we have that the ~T-operator subgroup R of sg0x �9 . . •162 is a homomorphism of l-I { ~ : v ~ / 3 into l - I { ~ : u~/3.

Now {0 . . . . . s}\I can have at most one element since ~[ is S.I. (If Uo, ua ~ / , uo ~ ux, then R = (R + O~o)f3 (R + Q,,), as follows from the above facts.) It must have at least one element since R ~ A o X . . . • So assume w.l.o.g, that I = { 1 . . . . , s}. Then it is quite easy to see that x~--~(x,O,O . . . . . 0)/~9 is an isomorphism between ~[o T and ~T.

To complete the proof, we make the following observation, whose easy proof is left to the reader.

L E M M A 16. A finite affine C.P. algebra ~ is simple iff it is para primal iff ~lv is simple. I f simple, ~ has no proper, non-trivial subalgebras.

T H E O R E M 17. s9 is directly representable by the set of its finite simple members, and this set has only finitely many non-isomorphic members.

Proof. By Theorems 11 and 15, every finite S.I. in ~ is simple and has cardinality not exceeding the maximum cardinality of ~1 . . . . , ~n- As we re- marked in the proof of (11.1), in a C.P. variety every subdirect product of a finite system of simple algebras is isomorphic to the product of some of them.

The above theorem implies Theorem 7 via Theorems 3 and 5. This done, we should now like to make a few further observations on para primal varieties. On the one hand, a finite simple affine C.P. algebra is para primal, and the study of para primal varieties generated by a finite set of these (affine varieties) is rather trivial and reduces in an obvious fashion to applications of the theory of modules over a finite ring. At the opposite pole from the affine algebras are the quasi primal algebras�9

DEFINITION 18. (1) An algebra ~ is quasi primal iff it is finite and the ternary discriminator function, defined by t(x, y, z) = x if x ~ y and t(x, x, z) = z, is a polynomial in g .

(2) A variety is quasi primal if[ it is C.P. and generated by a finite set of quasi primal algebras�9

For comparison with our following results, we recall a theorem of Pixley.

Vol 8, 1978 Para primal varieties 345

T H E O R E M 19 ([12; Thin. 1.2]). The following are equivalent, for a finite algebra ~[:

(1) ~ is quasi primal. (2) ~[ is para primal and ~(~) is congruence-distributive. (3) Every finitary operation on the universe of ~ which respects subalgebras of

and isomorphisms between subalgebras is a polynomial in ~.

Note that if ~ is C.P. because a certain term t satisfies the Mal'cev identities, and also ~ is quasi primal, it does not follow that t is the ternary discriminator in ~.

L E M M A 20. No congruence distributive variety has a non-trivial affine algebra.

T H E O R E M 21. The following are equivalent, for a para primal variety ~F:

(1) 7/" is quasi primal. (2) ~ is congruence distributive. (3) 7/" has no non-trivial affine algebras.

T H E O R E M 22. A finite algebra is quasi primal iff it is para primal and has no non-trivial affine subalgebras.

Proof. For 20 we use Jonsson's characterization of congruence distributivity ([5]). Suppose that ~ is affine over ~ and that to . . . . . tK are polynomials in satisfying to(X, y, z) = x, t~(x, y, z) = z, tn(x, y, x) = x and t2i(x, x, y) = t21§ x, y), t2~_~(x, y, y) = t2i(x, y, y) for n, 2 i+1 , 2]--- ~. Each tn can be written as tn(x, y, z) = a,(X)+[3n(y)+Tn(Z)+Cn where an, /3,, 3'n are endomorphisms of ~ . Now 0 = tn(0, 0, 0) = cn. Then 0 = tn(0, y, 0) =/3n (y). Thus tn(x, y, z) = t,(x, x, z) = t,(x, z, z )= %(x)+~/n(z). And it follows that ~ satisfies the identity x ~ z.

For 21: (1 ) f f (3). By 19 and 20, the quasi primal algebras generating ~" have no non-trivial affine subalgebras. Hence, by 12, ~ has none. ( 3 ) ~ (2). In Theorem 11.1 ~* disappears. Hence by 11.2 the congruence lattice of any finite

~ 7 r is distributive. Since ~ is locally finite this extends to all of ~r. (2) ~ (1). By Theorem 19.

For 22". ~ is by 19 and 20. If ~ is para primal with no affine non-trivial subalgebras, then by 12, ~(~) has none, so by 21 it is congruence distributive and by 19, we have that ~ is quasi primal.

Concluding remarks. Affine and quasi primal varieties are the two extremes among para primal varieties. Every para primal variety is, in the sense that

346 RALFH MCKENZIE ALGEBRA UNIV.

Theorem 11 is valid, quasi primal-by-affine and thus inherits the nice properties that these two kinds of variety have in common.

3. S o m e limitations: lattices

Varieties generated by a finite lattice with or without additional operations are residually * finite, congruence distributive and finitely axiomatizable. We may ask whether Theorem 3 can be used to provide an easy direct proof for the finite axiomatizability of such varieties. For most familiar sorts of lattices with operators, in particular relation algebras and finite dimensional cylindric algebras, the answer is clearly yes. Yet the following result shows that our Theorem 3 is not strong enough to derive the finite axiomatizability of varieties generated by a finite lattice. The question whether varieties generated by a finite group or ring have definable principal congruences is open.

T H E O R E M 23. The only varieties of lattices having definable principal con- gruences are the two varieties of distributive lattices (all distributive lattices, all 1-element lattices).

Proof. The two smallest non-distributive lattice varieties are those generated by the "d iamond" and the "pentagon". (See fig. 1.). Thus it suffices to show that neither of these has definable principal congruences. In the cartesian power "M3, let P, be the sublattice consisting of all functions f such that: i < ] < n implies i~-----fj, and I f - l{bI I - - -1 , If-l/c/I 1. Let 1, 0, a~ P, be the constant functions with values 1, 0, a, respectively. In the lattice "Ns, let Q, be the sublattice consisting of all functions f satisfying: i < ] < n implies i~-----f~, and I f - l M I _ 1. Let 1, b, a be the constants.

The reader can easily see that P,, Q, are indeed lattices. Moreover, in P,, (0, a) ~ 0(a, 1), and in Q,, (a, b) ~ 0(b, 1). This is because, for example, lattices being congruence distributive and P, being a subdirect power of M3, a necessary and sufficient condition for (u, v)~ 0(x, y) is that for all i < n, (hi, v~)~ 0M~(x, y~).

1 1

a c a<

o o

Figure 1

Vol 8, 1978 Para primal varieties 347

Let q , , ( x , , . . . , x2,,,, y) be the lattice polynomials defined recursively by:

qo(Y) = Y, qm+x(x, X2m+l , Xzm+2, Y) = X2m+2"}- X2rn+l " qm(~, y).

Let ~b,,(u, v, x, y) be the formula:

3Xo, �9 . �9 , x,,-11u " v~q,,,(Y~o,X" y ) A u + v

~ q,,($,,_~, x + y) A A i<rn--1

q,,,(.~, x + y ) ~ q~(.~+l, X " y)] .

Now it is known that in any lattice, ~b,, ~ (~rn+l for all m and

(u, v ) e O(x, y)r V 4~,,(u, v, x, y). m

Thus if a lattice variety has definable principal congruences, then one of the formulas ~b,, does the job.

Therefore, the following claim establishes our theorem.

Claim 1. Pn+l ~ &,(0, a, a, 1), On+l / ~bn(a, b, b, 1). This follows from the fol- lowing claims. Claim 2: In Pn+l, if 0<-q~(t ,a)_<q,(f , 1)_<a (any K-----l), then q~(f, a), q~(f, 1) differ at at most one j < n + t. Claim 3: In Qn+l, if a_< q,([, b) -< q,(f , 1)-<b, then qK(f, b), q,(f, 1) differ at at most one j < n + l . (To see that (2), (3) imply (1), note that if qb~(u , v , x , y ) holds (where u<-v, and x-<y) by Xo . . . . . x,~-i then u<-q,,,(.~, x)<-q,,,(Y,~, y)-<v for all i.)

Since the proofs of claims (2), (3) are analogous, we shall only do (3). Let i < j < n + 1. By induction on K, using the definition of Q~§ one easily shows that for any pair of values g = q , ( [ , b ) , h = q , ( f , 1), that the quadruple (g(i), g(j), h(i), h(j)) can only be one of the following, unless g( i )=h( i ) or g(])= h(]) : (b, b, 1, 1), (0, b, c, 1), (a, b, 1, 1), (a, b, b, 1). The conclusion follows.

We should remark that in distributive lattices

~b(u, v, x, y ) : u + x + y ~ v + x + y A u . x �9 y.-~v �9 x �9 y

defines the principal congruences. The variety of commutative idempotent semig- roups is an example of a residually * finite variety with definable principal congruences that satisfies no non-trivial congruence identities.

348 RALPH MCKENZIE

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University of California Berkeley, California U.S.A.