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

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion Let G be a group. We show that the Birget–Rhodes prefix expansion $$G^{Pr}$$ G Pr and the Margolis–Meakin expansion M(X; f) of G with respect to $$f:X\rightarrow G$$ f : X → 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 $$U/\zeta$$ U / ζ which is a homomorphic image of the free product U of the free semigroup $$X^+$$ 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 $$S/\sigma _S=G$$ S / σ S = G , from which the characterizing universal properties of $$G^{Pr}$$ G Pr and M(X; f) can be recaptured easily. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semigroup Forum Springer Journals

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

, Volume 96 (3) – Mar 22, 2018
16 pages

/lp/springer_journal/a-unifying-approach-to-the-margolis-meakin-and-birget-rhodes-group-libpUtqgdM
Publisher
Springer Journals
Subject
Mathematics; Algebra
ISSN
0037-1912
eISSN
1432-2137
D.O.I.
10.1007/s00233-018-9932-7
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Semigroup ForumSpringer Journals

Published: Mar 22, 2018

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