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U.W. Bangor - School of Informatics - Mathematics Preprints 1995

Semigroup and Automata Theory

95.07 : LAWSON, M.V.

Constructing Inverse Semigroups from Category Actions


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.

95.13 : HINES, P.M. & LAWSON, M.V.

An application of polycyclic monoids to rings


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.

95.14 : J.B. Hickey & M.V.Lawson

Unit regular monoids


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.

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