# Semigroup and Automata Theory

### Constructing Inverse Semigroups from Category Actions

#### Abstract:

The theory in this paper was motivated by an example of an inverse semigroup important in Girard's Geometry of Interaction programme for linear logic. At one level, the theory is a refinement of the Wagner-Preston representation theorem: we show that every inverse semigroup is isomorphis to an inverse semigroup of all partial symmetries (of a specific type) of some structure. At another level, the theory unifies and completes two classical theories: the theory of bisimple inverse monoids created by Clifford and subsequently generalised to all inverse monoids by Leech; and the theory on 0-bisimple inverse semigroups due to Reilly and McAlister. Leech showed that inverse monoids could be described by means of a class of right cancellative categories, whereas Reilly and McAlister showed that 0-bisimple inverse semigroups could be described by means of generalised RP-systems. In this paper, we prove that every inverse semigroup can be constructed from a category acting on a set satisfying what we term the `orbit condition'.

#### Published in:

Journal of Pure and Applied Algebra 137 (1999) 57-101.

### An application of polycyclic monoids to rings

#### Abstract:

An important technique in $K$-theory of certain $C^*$-algebras is that of "halving projections". The existence of such projections in a $C^*$-algebra $A$ leads to an isomorphism between $A$ and the ring of all $n \times n$ matrices over $A$.
This technique is also employed by Girard in his work on linear logic.
In this note, we show that such a process is equivalent to the existence of a 'strong embedding' of the polycyclic monoid on two generators.
We also provide a semigroup theoretic analogue of this result.
We shall assume that the reader is familiar with the elementary theory of inverse semigroups.

#### Published in:

Semigroup Forum 56 (1998) 146-149.

### Unit regular monoids

#### Abstract:

We derive necessary and sufficient conditions for a unit regular monoid to have a uniquely unit regular, idempotent separating cover.

#### Published in:

Proceedings of the Royal Society of Edinburgh 127A (1997) 127-144.