Partial group actions and their generalisations

The notion of the partial action of a group on a set first arose in the context of operator algebras and has since found a use in a wide range of other areas: model theory and tilings, for example. One question which has received particular attention is the following: given a partial group action, (when) can one construct an action? In this seminar, I will begin by discussing the definition of a partial group action (with examples), before describing one such method for constructing an action from a partial action: the process of `globalisation'. This is a construction which appears in a number of other contexts (topology, combinatorial group theory, ...). With the desired results established in the group case, I will then consider the generalisations to the cases of partial actions of monoids and of so-called `weakly left E-ample semigroups'. These latter semigroups arise naturally as subsemigroups of partial transformation monoids which are closed under a certain unary operation. We will see that the notion of `globalisation' must be modified if we are to obtain such results for the partial actions of weakly left E-ample semigroups.