Actions and expansions of ordered groupoids

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 Γ(G,A).