The Weak Extension Property and Finite Axiomatizability for Quasivarieties

We define and compare a selection of congruence properties of quasivarieties, including the relative congruence meet semi-distributivity, RSD(∧), and the weak extension property, WEP. We prove that if K ⊆ L ⊆ L′ are quasivarieties of finite signature, and L′ is finitely generated while K |= WEP, then K is finitely axiomatizable relative to L. We prove for any quasivariety K that K |= RSD(∧) iff K has pseudo-complemented congruence lattices and K |= WEP. Applying these results and other results proved in M. Maróti, R. McKenzie [17] we prove that a finitely generated quasivariety of finite signature L is finitely axiomatizable provided that L satisfies RSD(∧), or that L is relatively congruence modular and is included in a residually small congruence modular variety. This yields as a corollary the full version of R. Willard’s theorem for quasivarieties and partially proves a conjecture of D. Pigozzi. Finally, we provide a quasi-Maltsev type characterization for RSD(∧) quasivarieties and supply an algorithm for recognizing when the quasivariety generated by a finite set of finite algebras satisfies RSD(∧).