Equational closure operator and forbidden semidistributive lattices
Let \({\mathcal S}\) be a quasivariety. A subclass \({\mathcal R}\subset {\mathcal S}\) is called an \({\mathcal S}\)-variety if \({\mathcal R}={\mathcal S}\cap {\mathcal V}\) for some variety \({\mathcal V}\). Denote by \(L_ q({\mathcal S})\) the lattice of all subquasivarieties of \({\mathcal S}\). Let Q be the class of all lattices of quasivarieties. The author found some new property of Q-lattices which implies the known Dziobiak's property. It is shown that the operator on \(L_ q({\mathcal S})\) assigning to each subquasivariety its equational closure in \({\mathcal S}\) can be axiomatized and that every coalgebraic lattice is isomorphic to the lattice of \({\mathcal S}\)-varieties of some quasivariety S. The authors characterize ordered sets whose lattices of convex subsets are Q-lattices.
scientific article; zbMATH DE number 4168754