The weak extension property and finite axiomatizability forquasivarieties

Wiesław Dziobiak ¹   ; Miklós Maróti ²   ; Ralph McKenzie ³   ; Anvar Nurakunov ⁴

¹ Department of Mathematics University of Puerto Rico Mayagüez Campus Mayagüez, PR 00681-9018, U.S.A.
² Bolyai Institute University of Szeged H-6720 Szeged, Hungary
³ Department of Mathematics Vanderbilt University Nashville, TN 37235, U.S.A.
⁴ Institute of Mathematics National Academy of Science Chui pr., 265a Bishkek, 720071, Kyrghyz Republic

Fundamenta Mathematicae, Tome 202 (2009) no. 3, pp. 199-223

Résumé

We define and compare a selection of congruence properties of quasivarieties, including the relative congruence meet semi-distributivity, ${\rm RSD}(\wedge)$, and the weak extension property, ${\rm WEP}$. We prove that if ${{{\cal K}}}\subseteq {{{\cal L}}}\subseteq {{{\cal L}}}'$ are quasivarieties of finite signature, and ${{{\cal L}}}'$ is finitely generated while ${{{\cal K}}}\models {\rm WEP}$, then ${{{\cal K}}}$ is finitely axiomatizable relative to ${{{\cal L}}}$. We prove for any quasivariety ${{{\cal K}}}$ that ${{{\cal K}}}\models {\rm RSD}(\wedge)$ iff ${{{\cal K}}}$ has pseudo-complemented congruence lattices and ${{{\cal K}}}\models {\rm WEP}$. Applying these results and other results proved by M.~Maróti and R.~McKenzie [Studia Logica 78 (2004)] we prove that a finitely generated quasivariety ${{{\cal L}}}$ of finite signature is finitely axiomatizable provided that ${{{\cal L}}}$ satisfies ${\rm RSD}(\wedge)$, or that ${{{\cal L}}}$ 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 ${\rm RSD}(\wedge)$ quasivarieties and supply an algorithm for recognizing when the quasivariety generated by a finite set of finite algebras satisfies ${\rm RSD}(\wedge)$.

Détail

DOI : 10.4064/fm202-3-1

Keywords: define compare selection congruence properties quasivarieties including relative congruence meet semi distributivity rsd wedge weak extension property wep prove cal subseteq cal subseteq cal quasivarieties finite signature cal finitely generated while cal models wep cal finitely axiomatizable relative cal prove quasivariety cal cal models rsd wedge cal has pseudo complemented congruence lattices cal models wep applying these results other results proved mar mckenzie studia logica prove finitely generated quasivariety cal finite signature finitely axiomatizable provided cal satisfies rsd wedge cal relatively congruence modular included residually small congruence modular variety yields corollary full version willards theorem quasivarieties partially proves conjecture pigozzi finally provide quasi maltsev type characterization rsd wedge quasivarieties supply algorithm recognizing quasivariety generated finite set finite algebras satisfies rsd wedge