Finite Axiomatizability of Congruence Rich Varieties

In this paper we introduce the notion of a congruence rich variety of algebras, and investigate which locally finite subvarieties of such a variety are relatively finitely based. We apply the results obtained to investigate the finite axiomatizability of an interesting variety generated by a particular five-element directoid. Roughly speaking, congruence rich varieties are those in which all large finitely generated algebras have homomorphic images of moderate size. More precisely, a variety V is congruence rich provided for each positive integer n, there is a positive integer m such that every finitely generated algebra in V with more than n elements has a homomorphic image with more than n elements but no more than m elements. In this paper we restrict our attention to varieties of finite similarity types—that is, the algebras in the varieties we investigate are always assumed to have only finitely many basic operations. Among varieties of finite similarity type, congruence rich varieties are not uncommon. Any finitely generated congruence modular variety is congruence rich, as is any locally finite variety with a finite upper bound on the cardinalities of its finite subdirectly irreducible algebras. Any variety of directoids, which were introduced in [9] as algebraic renderings of up-directed sets, is congruence rich. Below we introduce varieties orderable-by-divisibility. These are also congruence rich, and the variety of directoids is a special case. Let A be an algebra. By an HS-reduct of A we mean any algebra in HS(A), i.e., any algebra which is a homomorphic image of a subalgebra of A. By a proper HS-reduct of A we mean one that is not isomorphic to A itself. In case A is a finite algebra, its proper HS-reducts are those with cardinality smaller than the cardinality of A. Let U be a variety. A finite algebra A is said to be critical for the variety U provided A does not belong to U but all of the proper HS-reducts of A belong to U . A is called critical provided it is critical for some variety. Evidently, every critical algebra is subdirectly irreducible. Critical groups have played a role in the theory of varieties of groups. The understanding of algebras critical for locally finite subvarieties of congruence rich varieties is a key to our results. In Section 1 we present several conditions which characterize congruence richness. We provide, for each locally finite subvariety of a congruence rich variety, a “forbidden HS-reduct” characterization in terms of critical algebras. We also present some examples of congruence rich varieties and some results on how to obtain new congruence rich varieties from those already at hand. In Section 2 we focus on finite axiomatizability. We give a necessary and sufficient condition for a locally finite subvariety U of a congruence rich variety V to be finitely based relative to V. Perhaps the most useful formulation is that only finitely many finite algebras, up to isomorphisn, in V should be critical for U . Likewise, we characterize those locally finite subvarieties which are inherently nonfinitely based relative to a congruence rich variety. This characterization also depends on critical algebras. Understanding finite axiomatizability of locally finite varieties, or even of varieties generated by a finite algebra, has proven to be very challenging. Roger Lyndon, in [12] gave the earliest example