The weak extension property and finite axiomatizability for quasivarieties

Wiesław Dziobiak, et al. "The weak extension property and finite axiomatizability for quasivarieties." Fundamenta Mathematicae 202.3 (2009): 199-223. <http://eudml.org/doc/283240>.

@article{WiesławDziobiak2009,
abstract = {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(∧).},
author = {Wiesław Dziobiak, Miklós Maróti, Ralph McKenzie, Anvar Nurakunov},
journal = {Fundamenta Mathematicae},
keywords = {quasivariety; finite axiomatizability; congruence properties},
language = {eng},
number = {3},
pages = {199-223},
title = {The weak extension property and finite axiomatizability for quasivarieties},
url = {http://eudml.org/doc/283240},
volume = {202},
year = {2009},
}

TY - JOUR
AU - Wiesław Dziobiak
AU - Miklós Maróti
AU - Ralph McKenzie
AU - Anvar Nurakunov
TI - The weak extension property and finite axiomatizability for quasivarieties
JO - Fundamenta Mathematicae
PY - 2009
VL - 202
IS - 3
SP - 199
EP - 223
AB - 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(∧).
LA - eng
KW - quasivariety; finite axiomatizability; congruence properties
UR - http://eudml.org/doc/283240
ER -