A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

Abstract

© 2018 Springer Science+Business Media, LLC, part of Springer Nature Let G be a group. We show that the Birget–Rhodes prefix expansion (Formula presented.) and the Margolis–Meakin expansion M(X; f) of G with respect to (Formula presented.) can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid (Formula presented.) which is a homomorphic image of the free product U of the free semigroup (Formula presented.) on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with (Formula presented.), from which the characterizing universal properties of (Formula presented.) and M(X; f) can be recaptured easily.