GRAPH EXPANSIONS OF RIGHT CANCELLATIVE MONOIDS

Abstract

The relations ℛ* and [Formula: see text] on a monoid M are natural generalizations of Green’s relations ℛ and [Formula: see text], which coincide with ℛ and [Formula: see text] if M is regular. A monoid M in which every ℛ*-class [Formula: see text] contains an idempotent is called left (right) abundant; if in addition the idempotents of M commute, that is, E(M) is a semilattice, then M is left (right) adequate. Regular monoids are obviously left (and right) abundant and inverse monoids are left (and right) adequate. Many of the well known results of regular and inverse semigroup theory have analogues for left abundant and left adequate monoids, or at least to special classes thereof. The aim of this paper is to develop a construction of left adequate monoids from the Cayley graph of a presentation of a right cancellative monoid, inspired by the construction of inverse monoids from group presentations, given by Margolis and Meakin in [10]. This technique yields in particular the free left ample (formerly left type A) monoid on a given set X.