Equations on the semídírect product of a finite semilattice by a -trivial monoid of height k,

Abstract

Abstract: Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q . The variety Com 1,1 is the variety of finite semilattices also denoted by J 1 . In this paper, we consider the product variety J 1 *Com t,q generated by all semidirect products of the form M * N with M J 1 and N Com t,q . We give a complete sequence of equations for J 1 * Com t,q implying complete sequences of equations for J 1 * (Com A), J 1 * (Com G) and J 1 * Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Article: 1. Introduction Let Com t , q denote the variety of finite monoids that satisfy the equations xy = yx and x t = x t+q . The variety Com 1,1 is the variety of finite semilattices also denoted by J 1 . In this paper, we give an equational characterization of the product variety J 1 * Com t , q generated by all semidirect products of the form M * N with M J 1 and N Com t , q . Our results imply a complete sequence of equations for J 1 * (Com A), J 1 *(Com G) and J 1 *Com, where Com denotes the variety of finite commutative monoids, A the variety of finite aperiodic monoids and G the variety of finite groups. Pin Our results follow from versions of techniques used in particular by , Brzozowski and Simon Definitions and notations Let M and N be monoids. We say that M divides N and write M N if M is a morphic image of a submonoid of N . Note that the divisibility relation is transitive. An M-variety V is a family of finite monoids that satisfies the following two conditions