Thesis (M. Sc.)--School of Mathematics and Statistics, Faculty of Science, University of Sydney, 2006. / Bibliography: leaves 53-54.

Thesis (M. Sc.)--University of Sydney, 2006. / Title from title screen (viewed 30 May 2008). Submitted in fulfilment of the requirements for the degree of Master of Science to the School of Mathematics and Statistics, Faculty of Science. Degree awarded 2006; thesis submitted 2005. Includes bibliographical references. Also available in print form.

Let S be a semigroup, C(S) the automaton constructed from the right Cayley graph of S with respect to all of S as the generating set and ∑(C(S)) the automaton semigroup constructed from C(S). Such semigroups are termed Cayley automaton semigroups. For a given semigroup S we aim to establish connections between S and ∑(C(S)). For a finite monogenic semigroup S with a non-trivial cyclic subgroup C[sub]n we show that ∑(C(S)) is a small extension of a free semigroup of rank n, and that in the case of a trivial subgroup ∑(C(S)) is finite. The notion of invariance is considered and we examine those semigroups S satisfying S ≅ ∑(C(S)). We classify which bands satisfy this, showing that they are those bands with faithful left-regular representations, but exhibit examples outwith this classification. In doing so we answer an open problem of Cain. Following this, we consider iterations of the construction and show that for any n there exists a semigroup where we can iterate the construction n times before reaching a semigroup satisfying S ≅ ∑(C(S)). We also give an example of a semigroup where repeated iteration never produces a semigroup satisfying S ≅ ∑(C(S)). Cayley automaton semigroups of infinite semigroups are also considered and we generalise and extend a result of Silva and Steinberg to cancellative semigroups. We also construct the Cayley automaton semigroup of the bicyclic monoid, showing in particular that it is not finitely generated.

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Much work has been done on the ℓ¹-algebras of groups, but much less on ℓ¹-algebras of semigroups. This thesis studies those of inverse semigroups, also known as generalised groups, with emphasis on the involutive structure. Where results extend to the semigroup ring, I extend them. I determine the characters of a semilattice in terms of its order structure. The simplest suffice to separate its ℓ¹-algebra. I also determine the algebra's minimal idempotents. I introduce a generalisation of Banach *-algebras which has good hereditary properties and includes the inverse semi groups rings. These latter have an ultimate identity which can be used to test for representability. Involutive semigroups with s*s an idempotent yield inverse semi groups when quotiented by the congruence induced by their algebras' *-radical. The left regular *-representation of inverse seroigroups is faithful and acts like that of groups. The corresponding idea of amenability coincides with the traditional one. Brandt semi groups have the weak containment property iff the associated group does. The relationship of ideals to weak containment is studied, and inverse semigroups with well ordered semilattices are shown to have the property if all their subgroups do. The converse is extended for Clifford semigroups. Symmetry and related ideas are considered, and basic results proved for the above mentioned generalisation, and a better version for a possibly more restricted generalisation. The symmetry of an ℓ¹-algebra of an E-unitary inverse semi group is shown to depend on the symmetry of the ℓ¹-algebra of its maximal group homomorphic image if the semilattice has a certain structure or the semigroup is a Clifford semigroup. Inverse semi groups with well ordered semilattices are shown to have symmetric ℓ¹-algebra if all the subgroups do. Finally, some topologically simple ℓ¹-algebras and simple semigroup rings are constructed, extending results on simple inverse semigroup rings.

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A double semigroup is a set equipped with two associative binary operations satisfying the middle-four interchange law. A double inverse semigroup is a double semigroup in which both operations are inverse semigroup operations. It is shown by Kock (2007) that all double inverse semigroups must be commutative. In this thesis, we define the notion of a double inductive groupoid which admits both a construction of double inverse semigroups given any double inductive groupoid, and vice-versa. These constructions are functorial and induce an isomorphism of categories between the category of double inductive groupoids with inductive functors and double inverse semigroups with double semigroup homomorphisms. By a further investigation of double inverse semigroups, we are able to show that the two operations of any double inverse semigroups must coincide and thus double inverse semigroups are commutative inverse semigroups.

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