Equations on the semidirect product of a finite semilattice by a finite commutative monoid

Abstract

LetCom t,q denote the variety of finite monoids that satisfy the equationsxy=yx andx t=x t+q. The varietyCom 1,1 is the variety of finite semilattices also denoted byJ 1. In this paper, we consider the product varietyJ 1*Com t,q generated by all semidirect products of the formM*N withM∈J 1 andN∈Com t,q . We give a complete sequence of equations forJ 1*Com t,q implying complete sequences of equations forJ 1*(Com∩A),J 1*(Com∩G) andJ 1*Com, whereCom denotes the variety of finite commutative monoids,A the variety of finite aperiodic monoids andG the variety of finite groups.