The weak extension property and finite axiomatizability for quasivarieties

• Autorzy: Wiesław Dziobiak , Miklós Maróti , Ralph McKenzie , Anvar Nurakunov
• Języki publikacji: EN

Abstrakty

EN

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(∧).

Kategorie tematyczne

• 08B10: Congruence modularity, congruence distributivity
• 08C15: Quasivarieties
• 03C05: Equational classes, universal algebra
• 08A99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2009
• Tom: 202
• Numer: 3
• Strony: 199-223

Daty

wydano

2009

Twórcy

autor

Wiesław Dziobiak

• Department of Mathematics, University of Puerto Rico, Mayagüez Campus, Mayagüez, PR 00681-9018, U.S.A.

autor

Miklós Maróti

• Bolyai Institute, University of Szeged, H-6720 Szeged, Hungary

autor

Ralph McKenzie

• Department of Mathematics, Vanderbilt University, Nashville, TN 37235, U.S.A.

autor

Anvar Nurakunov

• Institute of Mathematics, National Academy of Science, Chui pr., 265a, Bishkek, 720071, Kyrghyz Republic

Identyfikatory

DOI

10.4064/fm202-3-1