On the Varieties of Languages Associated with Some Varieties of Finite Monoids with Commuting Idempotents

Eilenberg has shown that there is a one-to-one correspondence between varieties of finite monoids and varieties of recognizable languages. In this paper, we give a description of a variety of languages close to the class of piecewise testable languages considered by I. Simon. The corresponding variety of monoids is the variety of J -trivial monoids with commuting idempotents. This result is then generalized to the case of finite monoids with commuting idempotents whose regular D-classes are groups from a given variety of groups.