Topic
Bicyclic semigroup
About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.
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TL;DR: For a semigroup S and a set B, the relative rank of S modulo A is the minimal cardinality of a setB such that S can be generated.
Abstract: For a semigroup S and a set the relative rank of S modulo A is the minimal cardinality of a setB such that generates S. We show that the relative rank of an infinite full transformation semigroup modulo the symmetric group, and also modulo the set of all idempotent mappings, is equal to 2. We also characterise all pairs of mappings which, together with the symmetric group or the set of all idempotents, generate the full transformation semigroup.
70 citations
01 Jun 2008
TL;DR: In this article, a topological groupoid is constructed from an inverse semigroup and shown to be isomorphic to the universal groupoid introduced by Paterson, and a certain reduction of this groupoid was proved to be a graph groupoid.
Abstract: We show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced -algebra.
70 citations
68 citations
65 citations
Journal Article•
TL;DR: A survey of the history of the study of weakly left Eample semigroups can be found in this paper, where the basic aspects of their theory are sketched out.
Abstract: Left restriction semigroups are a class of semigroups which generalise inverse semigroups and which emerge very naturally from the study of partial transformations of a set. Consequently, they have arisen in a variety of different contexts, under a range of names. One of the various guises under which left restriction semigroups have appeared is that of weakly left E-ample semigroups, as studied by Fountain, Gomes, Gould and Lawson, amongst others. In the present article, we will survey the historical development of the study of left restriction semigroups, from the `weakly left E-ample' perspective, and sketch out the basic aspects of their theory.
62 citations