A Finite Basis Theorem for Difference-term Varieties with a Finite Residual Bound

We prove that if V is a variety of algebras (i.e., an equationally axiomatizable class of algebraic structures) in a finite language, V has a difference term, and V has a finite residual bound, then V is finitely axiomatizable. This provides a common generalization of R. McKenzie’s finite basis theorem for congruence modular varieties with a finite residual bound, and R. Willard’s finite basis theorem for congruence meet-semidistributive varieties with a finite residual bound. This paper is a contribution to an old problem in logic from the schools of A. Tarski and A. Maltsev: which finite algebraic structures A (algebras for short) have a finite basis for their identities? Equivalently, for which finite algebras A is the variety V(A) (the smallest equational class containing A) finitely axiomatizable? On the one hand, every finite group [38], finite ring [22, 28], finite commutative semigroup [40], finite lattice [33], or two-element algebra in a finite language [30] is known to be finitely based. On the other hand, the list of finite algebras which are not finitely based includes, in addition to pathological examples (e.g., [31, 37]), some finite semigroups [40], some finite non-associative K-algebras [41, 29, 12], and even a finite group with one non-identity element named by a constant [5]. In 1996 R. McKenzie [36] proved that the problem of determining whether a finite algebra is finitely based is undecidable, settling Tarski’s finite basis problem. The evidence suggests that a full classification of finitely based finite algebras is beyond reach. However, there are some remarkable partial results. In particular, in the early 1970s K. Baker [1, 2] proved the following: if A is a finite algebra in a finite language and V(A) is congruence distributive (i.e., for every B ∈ V(A), the lattice of congruence relations of B is a distributive lattice), then A is finitely based. Two important ingredients in the proof were provided by B. Jónsson [13]: (1) a characterization, in terms of identities, of the condition that a variety be congruence distributive, and (2) a proof that if A is finite and V(A) is congruence distributive, then every subdirectly Date: July 8, 2014. 2010 Mathematics Subject Classification. Primary 03C05; Secondary 08B05.