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 by M. Maróti and R. McKenzie [Studia Logica 78 (2004)] we prove that a finitely generated quasivariety ℒ of finite signature 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(∧).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm202-3-1,
author = {Wies\l aw Dziobiak and Mikl\'os Mar\'oti and Ralph McKenzie and Anvar Nurakunov},
title = {The weak extension property and finite axiomatizability for quasivarieties},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {199-223},
zbl = {1170.08004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm202-3-1}
}
Wiesław Dziobiak; Miklós Maróti; Ralph McKenzie; Anvar Nurakunov. The weak extension property and finite axiomatizability for quasivarieties. Fundamenta Mathematicae, Tome 205 (2009) pp. 199-223. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm202-3-1/