TOPOLOGICAL PROPERTIES OF A QUASIVARIETY GENERATED BY A SPECIAL MODULAR LATTICE

Abstract

In universal algebra and lattice theory, one of the central questions is the standardness problem for quasivarieties of algebras. This problem, formulated by D.M. Clark, B.A. Davey, M. Jackson and J. Pitkethly in 2008, asks: which finite lattices generate standard topological quasivarieties? This problem is closely related to the question of finite axiomatizability of lattices, namely: which finite lattices have a finite basis of quasi-identities?The formulation of this question goes back to the work of V.A. Gorbunov and D.M. Smirnov in 1979. In this work, sufficient conditions are established under which a finite modular lattice generates a non-finitely axiomatizable and non-standard topological quasivariety. Corresponding examples demonstrating the realizability of these conditions are constructed, and a generalization is obtained that confirms the existence of an infinite family of such lattices.