Actions and expansions of ordered groupoids

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18 Citations (Scopus)

Abstract

We construct the Birget-Rhodes expansion GBR of an ordered groupoid G. The construction is given in terms of certain finite subsets of G, but we also show how GBR can be considered as a prefix expansion. Moreover, for an inductive groupoid G we recover the prefix expansion of Lawson-Margolis-Steinberg. We show that the Birget-Rhodes expansion of an ordered groupoid G classifies partial actions of G on a set X: the correspondence between partial actions of G and actions of GBR can be viewed as a partial-to-global result achieved by enlarging the acting groupoid. We further discuss globalisation achieved by enlarging the set acted upon and show that a groupoid variant of the tensor product of G-sets provides a canonical globalisation of any partial action. We also sketch the construction of the Margolis-Meakin expansion (G, A)MM of an ordered groupoid G with generating set A. This is related to the Birget-Rhodes expansion and was first defined by Margolis and Meakin for a group G generated by A in terms of finite subgraphs of the Cayley graph G(G, A). © 2004 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)175-195
Number of pages21
JournalJournal of Pure and Applied Algebra
Volume198
Issue number1-3
DOIs
Publication statusPublished - 1 Jun 2005

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