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

Abstract

Let G be a group. We show that the Birget–Rhodes prefix expansion $$G^{Pr}$$ and the Margolis–Meakin expansion M(X; f) of G with respect to $$f:X\rightarrow G$$ 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 $$U/\zeta $$ which is a homomorphic image of the free product U of the free semigroup $$X^+$$ 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 $$S/\sigma _S=G$$ , from which the characterizing universal properties of $$G^{Pr}$$ and M(X; f) can be recaptured easily.