Actions and expansions of ordered groupoids

Abstract

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 Γ ( G , A ) .