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:

• The filter in L generated by Q(F) is prime.

• For every quasivariety ${\mathcal {K}}$ contained in ${\mathcal {M}}^{-}_{3}$, the interval $[{\mathcal {K}}, {\mathcal {V}}]$ contains an isomorphic copy of the ideal lattice I(F) of F. In particular, the filter in L generated by Q(F) contains an isomorphic copy of I(F).

• The distributive lattice of the order ideals of $({\mathcal {S}}, \le )$, where $A \le B$ means $A \in Q(B)$, is uncountable and is a homomorphic image of the Q-lattice of Q(Fin).