, Volume 78, Issue 1-2, pp 293-320

Finite basis problems and results for quasivarieties

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Abstract

Let \(\mathcal{K}\) be a finite collection of finite algebras of finite signature such that SP(\(\mathcal{K}\)) has meet semi-distributive congruence lattices. We prove that there exists a finite collection \(\mathcal{K}\) 1 of finite algebras of the same signature, \(\mathcal{K}_1 \supseteq \mathcal{K}\), such that SP(\(\mathcal{K}\) 1) is finitely axiomatizable.We show also that if \(HS(\mathcal{K}) \subseteq SP(\mathcal{K})\), then SP(\(\mathcal{K}\) 1) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.