Existence of independent quasi-equational bases
The paper belongs to a series of papers of the author and her collaborators concerning independent quasi-equational bases for quasivarieties of algebraic structures, and the related problem of complexity of lattices of quasivarieties. Let \(\mathbf{K} \subseteq \mathbf{M}\) be classes of algebraic structures of the same similarity type \(\sigma\). As defined by the author, the class \(\mathbf{K}\) has an \emph{independent quasi-equational basis relative to} \(\mathbf{M}\), if \(\mathbf{K} = \mathbf{M} \cap \mathrm{Mod}(\Phi)\) for a set \(\Phi\) of quasi-identities and \(\mathbf{K} \neq \mathbf{M} \cap \mathrm{Mod}(\Phi \setminus \{\varphi\})\) for each \(\varphi \in \Phi\). A quasivariety \(\mathbf{K}\) has an \emph{independent quasi-equational basis} if \(\mathbf{K}\) has an independent quasi-equational basis relative to the class of all structures of type \(\sigma\). A sufficient condition is given for a quasivariety \(\mathbf{K}\) which ensures that \(\mathbf{K}\) contains continuum many subquasivarieties with no independent quasi-equational basis relative to \(\mathbf{K}\). This result is a weaker version of an earlier result given by A. V. Kravchenko, A. M. Nurakunov and the author. Some interesting consequences and applications of the author's result are provided. In particular, the condition holds for any so-called almost \(\mathit{f \, f}\)-universal quasivariety \(\mathbf{K}\). Also some further open problems are formulated.
scientific article; zbMATH DE number 7226858
scientific article