An amalgam of inverse semigroups is embedded into an amalgam with a lower bounded core

Abstract

Given any amalgam [Formula: see text] of inverse semigroups, we show how to construct an amalgam [Formula: see text] such that [Formula: see text] is embedded into [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and, for any [Formula: see text] and [Formula: see text] with [Formula: see text] in [Formula: see text], where [Formula: see text], there exists [Formula: see text] with [Formula: see text] in [Formula: see text]; that is, [Formula: see text] is a lower bounded subsemigroup of [Formula: see text] and [Formula: see text]. A recent paper by the author describes the Schützenberger automata of [Formula: see text], for an amalgam [Formula: see text] where [Formula: see text] is lower bounded in [Formula: see text] and [Formula: see text], giving conditions for [Formula: see text] to have decidable word problem. Thus we can study [Formula: see text] by considering [Formula: see text]. As an example, we generalize results by Cherubini, Jajcayová, Meakin, Piochi and Rodaro on amalgams of finite inverse semigroups.