A group G is said to act partially on a set Y if there is a map : G ! I(Y ) into the semigroup of partial bijections on X such that (g)(h) (gh) and (g) = (g ) for all g; h 2 G. We prove that each partial group action is the restriction of a universal global group action, and show that this result provides the key to understanding Munn's proof of the P -theorem.

McAlister’s P-theorem for E–unitary inverse semigroups is one of the most significant components of the structure theory for inverse semigroups: since its first appearance in 1974 several different proofs have been given and the scope of the theorem has been extended to strongly E ∗ –unitary and strongly categorical inverse semigroups. In this paper, we prove a P –theorem for the wider class of incompressible ordered groupoids, which encompasses previous versions of the P –theorem and offers a unified proof. Moreover, the class of incompressible ordered groupoids may be of interest in its own right, and we look at examples related to Bass-Serre theory for groups. 1

We generalise the Margolis-Meakin graph expansion of a group to a construction for ordered groupoids, and show that the graph expansion of an ordered groupoid enjoys structural properties analogous to those for graph expansions of groups. We also use the Cayley graph of an ordered groupoid to prove a version of McAlister’s P –theorem for incompressible ordered groupoids.

Abstract. Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then 〈 G, a 〉 \ G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates a g = g −1 ag of a by elements g ∈ G generate a semigroup denoted 〈a g | g ∈ G〉. We classify the finite permutation groups G on a finite set X such that the semigroups 〈G, a〉, 〈G, a〉\G, and 〈a g | g ∈ G 〉 are regular for all transformations of X. We also classify the permutation groups G on a finite set X such that the semigroups 〈 G, a 〉 \ G and 〈 a g | g ∈ G 〉 are generated by their idempotents for all non-invertible transformations of X.