Inverse Semigroups Acting on Graphs

There has been much work done recently on the action of semigroups on sets with some important applications to, for example, the theory and structure of semigroup amalgams. It seems natural to consider the actions of semigroups on sets ‘with structure’ and in particular on graphs and trees. The theory of group actions has proved a powerful tool in combinatorial group theory and it is reasonable to expect that useful techniques in semigroup theory may be obtained by trying to ‘port’ the Bass-Serre theory to a semigroup context. Given the importance of transitivity in the group case, we believe that this can only reasonably be achieved by restricting our attention to the class of inverse semigroups. However, it very soon becomes apparent that there are some fundamental differences with inverse semigroup actions and even such basic notions such as free actions have to be treated carefully. We make a start on this topic in this paper by first of all recasting some of Schein’s work on representations by partial homomorphisms in terms of actions and then trying to ‘mimic’ some of the basic ideas from the group theory case. We hope to expand on this in a future paper [5].