On formations of monoids

Abstract

A formation of monoids is a class of finite monoids closed under taking quotients and subdirect products. Formations of monoids were first studied in connection with formal language theory, but in this paper, we come back to an algebraic point of view. We give two natural constructions of formations based on constraints on the minimal ideal and on the maximal subgroups of a monoid. Next we describe two sublattices of the lattice of all formations, and give, for each of them, an isomorphism with a known lattice of varieties of monoids. Finally, we study formations and varieties containing only Clifford monoids, completely describe such varieties and discuss the case of formations.