, 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.

While working on this paper, the first author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877 and the second author was supported by the US National Science Foundation grant no. DMS-0245622.
Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko