Actions and expansions of ordered groupoids

We construct the Birget-Rhodes expansion G^BR of an ordered groupoid G. The construction is given in terms of certain finite subsets of G, but we also show how G^BR 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 G^BR 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.