Algebraically Expandable Classes of Implication Algebras

In this work we solve the following problem: Characterize the subclasses of implication algebras that can be axiomatized by sentences of the form 89! ^ p = q. In the process we obtain a representation result for …nite implication algebras, and as a by-product of our solution a number of interesting classes of implication algebras arise. We also obtain a characterization of the congruence permutable implication algebras. Implication algebras, also known as Tarski algebras, have been introduced and studied by J. C. Abbott in [1], [2]. They are the f!g-subreducts of Boolean algebras. It is also known that implication algebras are the algebraic counterpart of the implicational fragment of classical propositional logic [4]. An implication algebra is an algebra (L;!; 1) satisfying: (I1) 1! x x; (I2) x! 1 1; (I3) x! (y ! z) y ! (x! z); (I4) (x! y)! y (y ! x)! x: We write I to denote the variety of implication algebras. The algebra 2 = (f0; 1g;!; 1), where x ! y = 0 i¤ x = 1 and y = 0, is the only (up to isomorphisms) subdirectly irreducible in I. An equational function de…nition sentence (EFD-sentence) is a sentence of the form 8x1; :::; xn9!z1; :::; zm "(~x; ~z),