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 language | English |
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Pages (from-to) | 175-195 |
Number of pages | 21 |
Journal | Journal of Pure and Applied Algebra |
Volume | 198 |
Issue number | 1-3 |
DOIs | |
Publication status | Published - 1 Jun 2005 |