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We consider proper (idempotent pure) extensions of weakly left ample semigroups. These are extensions that are injective in each \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\widetilde{\mathcal{R}}$ \end{document}-class. A graph expansion of a weakly left ample semigroup S is shown to be such an extension of S. Using semigroupoids acted upon by weakly left ample semigroups, we prove that any weakly left ample semigroup which is a proper extension of another such semigroup T is (2,1)-embeddable into a λ-semidirect product of a semilattice by T. Some known results, by O'Carroll, for idempotent pure extensions of inverse semigroups and, by Billhardt, for proper extensions of left ample semigroups follow from this more general situation.

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Authors: Nassraddin Ghroda and Victoria Gould

Abstract  

We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford’s seminal work describing bisimple inverse monoids in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford’s work to bisimple inverse semigroups (a step that has previously proved to be awkward). We also put some earlier work of Gantos into a wider and clearer context, and pave the way for further progress.

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