# Semigroup and Automata Theory

### E*-unitary inverse semigroups

#### Abstract:

With each inverse monoid with zero S we associate a category C(S) which is a slight modification of the category Leech associated with an inverse monoid and a special case of the author's category action approach' to inverse semigroups. Using this category, we obtain characterisations of E*-unitary, strongly E*-unitary and E-unitary inverse monoids. Specifically,
• S is E*-unitary if and and only if C(S) is cancellative;
• S is strongly E*-unitary if and only if C(S) can be embedded in a groupoid;
• S is E-unitary (with a zero adjoined) if and only if C(S) has a groupoid of fractions.
We also introduce the universal group of an inverse semigroup' and show how this group can be used to characterise strongly E*-unitary semigroups.

#### Published in:

Semigroups, algorithms, automata and languages (eds. G.M.S. Gomes, J-E. Pin & P.V. Silva), World Scientific (2002) 195-214.

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gzipped postscript file: 02_10.ps.gz

### Partial actions of groups

#### Abstract:

A partial action of a group G on a set X is a weakening of the usual notion of a group action: the function G \times X \to X that defines a group action is replaced by a partial function; in addition, the existence of g \cdot (h \cdot x) implies the existence of (gh) \cdot x, but not necessarily so conversely. Such partial actions are extremely widespread in mathematics, and the main aimm of this paper is to prove two basic results concerning them. First, we obtain an explicit description of Exel's universal semigroup S(G), whihc has the property that partial actions of the group G give rise to actions of the inverse semigroup S(G). We apply this result to the theory of graph immersions. Second, we prove that each partial group action is the restriction of a universal global group action. We describe some applications of this result with group theory and the theory of E-unitary inverse semigroups.

#### Published in:

Internat. J. Algebra Comput. 14 (2004) 87-114.

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gzipped postscript file: 02_34.ps.gz

### Expansions of inverse semigroups

#### Abstract:

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of E-unitary inverse semigroups.

#### Published in:

J Austral. Math. Soc. 80 (2006) 205-228.