On some properties of quasivarieties generated by specific finite modular lattices

Abstract

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