Remarks about the Q-lattice of the variety of lattices

Abstract

Let L denote the Q-lattice of the variety $${\mathcal {V}}$$ of lattices, i.e. the lattice of quasivarieties that are contained in $${\mathcal {V}}$$ . Let F denote the free lattice in $${\mathcal {V}}$$ with $$\omega $$ free generators and let Q(F) denote the quasivariety of lattices generated by F. Let Fin denote the collection of finite lattices which belong to Q(F) and let Q(Fin) denote the quasivariety generated by Fin. Moreover, let $${\mathcal {M}}^{-}_{3}$$ denote the quasivariety of lattices which do not contain an isomorphic copy of $$M_3$$ (the 5-element non-distributive modular lattice) as a sublattice and let $${\mathcal {S}}$$ denote a selector of non-isomorphic finite quasicritical lattices which belong to Q(Fin). In this paper, we establish the following: