Strongly finitely based equational theories

Abstract

Let ~ denote the result of discarding the replacement rules from the Birkhoff calculus [- for equational logic. ~ is not necessarily incomplete, i.e., for suitable sets F of equations F ]- e may imply F ~ e, for all equations e. Call an equational theory T strongly based on F __ Tiff each equation e in T is derivable from F in the rudimentary Birkhoff calculus ~, that is, F ~ e. T is said to be strongly finitely based (s.f.b.) provided T is strongly based on some finite F. Clearly, a s.f.b, theory isfb. (finitely based). Conversely, many familiar lb. equational theories turn out to be sf.b., others not. The calculus ~ looks artificial only at first glance. It has interesting applications in general propositional logic. ~ links equational logic with Hilbert style axiomatization in a clear way: A variety V (more precisely, its equation theory) is sf.b. iff the propositional consequence defined by all logical matrices whose underlaying algebras belong to V is finitely axiomatizable, i.e., based on finitely many inference rules (cf. [10], also for the mentioned applications). Roughly speaking, replacement, if translated into propositional logic, is not expressible by Hilbert style inference rules. It has a particular status. A referee of the present paper also suggested the application to computerized proof systems since in a s.f.b, theory the time spent for searching matches is greatly reduced. Section 1 contains some examples and counterexamples of sf.b. theories and a generalization of a well-known result from [7], saying that the regularization T ~ of a strongly irregular lb. equational theory T is again f.b. Theorem 1 states the corresponding for the s.f.b, case. This is one of the steps in getting our main result, Theorem 2, saying that each variety of groupoids generated by its proper 2-element members is s.f.b. (A groupoid (A, �9 ) is called proper here if. depends effectively on both arguments. There are five proper 2-element groupoids, up to isomorphism.) As