U.W. Bangor  School of Informatics
Mathematics Preprints 2002
Semigroup and Automata Theory
02.10 : LAWSON, M.V.
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
Eunitary 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 Eunitary (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, JE. Pin & P.V. Silva),
World Scientific (2002) 195214.
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02_10.ps.gz
02.34 : KELLENDONK, J. & LAWSON, M.V.
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
Eunitary inverse semigroups.
Published in:
Internat. J. Algebra Comput.
14 (2004) 87114.
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gzipped postscript file:
02_34.ps.gz
02.35 : LAWSON, M.V., MARGOLIS, S. & STEINBERG, B.
Expansions of inverse semigroups
Abstract:
We construct the freest idempotentpure expansion of an inverse semigroup,
generalizing an expansion of Margolis and Meakin for the group case.
We also generalize the BirgetRhodes 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 idempotentpure homomorphisms
which we term
Fmorphisms.
These play the same role in the theory of idempotentpure homomorphisms
that
Finverse monoids play in the theory of
Eunitary inverse semigroups.
Published in:
J Austral. Math. Soc.
80 (2006) 205228.
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