The weak extension property and finite axiomatizability forquasivarieties
Volume 202 / 2009
Abstract
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 ${\rm RSD}(\wedge)$.
