Inverse semigroups acting on graphs

Wareham, Mark, John (2022) Inverse semigroups acting on graphs. University of Southampton, Doctoral Thesis, 83pp.

Record type: Thesis (Doctoral)

Abstract

In group theory we are able to derive many properties about a group from how it acts on a graph. Knowing this, we aimed to find similar results for inverse semigroups acting on graphs. We were able to find a consistent method of defining an action for a free product of inverse semigroups provided we already have actions for the semigroups that make up this product. Furthermore, this action will deliver back to us a fundamental inverse semigroup that is isomorphic to the free product. Following this, we looked at how our method works with polycyclic, Bruck-Reilly and Brandt semigroups. After finding that it does not work in the general case, we looked at what additional properties we will need for our semigroup in order to make it work. In particular, we found that a zero element in an inverse semigroup causes a lot of problems for our current method.

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