On some properties of quasivarieties generated by specific finite modular lattices

Abstract

A finite algebra A with discrete topology generates a topological quasivariety consisting of all topologically closed subalgebras of non-zero direct powers of A endowed with the product topology. This topological quasivariety is standard if every Boolean topological algebra with the algebraic reduct in Q(A) is profinite. In the article it is constructed the specific finite modular lattice T that does not satisfy one of Tumanov’s conditions but quasivariety Q(T) generated by this lattice is not finitely based. We investigate the topological quasivariety generated by the lattice T and prove that it is not standard. And we also would like to note that there is an infinite number of lattices similar to the lattice T.