Graph-monoids are introduced as algebraic objects which correspond to congruences over graphs. Varieties are defined, and a one-to-one correspondence between varieties of graph-monoids and varieties of graph-congruences is demonstrated. This is viewed as a generalization of the known correspondence between monoid varieties and congruence varieties. Some classification and effective characterizations of graph-monoid varieties are obtained. These are used to decide membership in the wreath product of certain monoids varieties and the variety of definite semigroups.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Sept. 18, 2008). PDF text: 75 p. ; 408 K. UMI publication number: AAT 3303784. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph.D.)--University of Central Florida, 2008. / Advisers: Gary Richardson, Ram N. Mohapatra. Includes bibliographical references (p. 55-57).

Thesis (Ph. D.)--Ohio State University, 2009. / Title from first page of PDF file. Includes vita. Includes bibliographical references (p. 124-125).

Thesis (Ph.D.) - University of St Andrews, June 2010.

Thesis (M.A.)--Kutztown State College. / Source: Masters Abstracts International, Volume: 45-06, page: 3170. Bibliography: leaves 60-61.

Thesis (Ph. D.)--University of Sydney, 2007. / Title from title screen (viewed 19 January 2009). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics. Includes bibliographical references. Also available in print form.

In this thesis we start by considering conditions under which some standard semigroup constructions preserve automaticity. We first consider Rees matrix semigroups over a semigroup, which we call the base, and work on the following questions: (i) If the base is automatic is the Rees matrix semigroup automatic? (ii) If the Rees matrix semigroup is automatic must the base be automatic as well? We also consider similar questions for Bruck-Reilly extensions of monoids and wreath products of semigroups. Then we consider subsemigroups of free products of semigroups and we study conditions that guarantee them to be automatic. Finally we obtain a description of the subsemigroups of the bicyclic monoid that allow us to study some of their properties, which include finite generation, automaticity and finite presentability.

To provide a general framework for the theory of automatic groups and semigroups, we introduce the notion of an automatic semigroup act. This notion gives rise to a variety of definitions for automaticity depending on the set chosen as a semigroup act. Namely, we obtain the notions of automaticity, Schutzenberger automaticity, R- and L-class automaticity, etc. We discuss the basic properties of automatic semigroup acts. We show that if S is a semigroup with local right identities, then automaticty of a semigroup act is independent of the choice of both the generators of S and the generators of the semigroup act. We also discuss the equality problem of automatic semigroup acts. To give a geometric approach, we associate a directed labelled graph to each S-act and introduce the notion of the fellow traveller property in the associated graph. We verify that if S is a regular semigroup with finitely many idempotents, then Schutzenberger automaticity is characterized by the fellow traveller property of the Schutzenberger graph. We also verify that a Schutzenberger automatic regular semigroup with finitely many idempotents is finitely presented. We end Chapter 3 by proving that an inverse free product of Schutzenberger automatic inverse semigroups is Schutzenberger automatic. In Chapter 4, we first introduce the notion of finite generation and finite presentability with respect to a semigroup action. With the help of these concepts we give a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated and finitely presented as an inverse semigroup. We end Chapter 4 by giving a necessary and sufficient condition for the semidirect product of a semilattice by a group to be Schutzenberger automatic. Chapter 5 is devoted to the study of HNN extensions of inverse semigroups from finite generation and finite presentability point of view. Namely, we give necessary and sufficient conditions for finite presentability of Gilbert's and Yamamura's HNN extension of inverse semigroups. The majority of the results contained in Chapter 5 are the result of a joint work with N.D. Gilbert and N. Ruskuc.

Neste trabalho de dissertação de mestrado, fazemos algumas observações a respeito de dois¶ artigos de pesquisa recentes, os quais estabelecem resultados falsos sobre existência de soluções controlabilidade exata de sistemas de controle abstratos do tipo neutro modelados na forma¶8<:¶ d ¶ dt ¶(x(t) + G(t, xt)) = Ax(t) + F(t, xt) + Cu(t), t 2 I = (0, a],¶ x0 = ',¶ (4)¶ onde A é o gerador infinitesimal de um C0-semigrupo de operadores lineares definidos sobre um espaço de Banach X; a função xt representa a história do estado no tempo t; C : U ! X ´e um¶ perador linear limitado, U é um espaço de Banach e F,G são funções apropriadas.¶ Motivados pelo anterior, neste trabalho estudamos a existência de soluções fracas para o sistema ¶ abstrato 8<:¶ d¶ dt¶ [(I &#8722; B)x(t)] = Ax(t) + F(t, x(t)), t 2 I = (0, a],¶ x(0) = x0 2 X,¶ (5)¶ onde B : X ! X é um operador linear limitado. Além do anterior, introduzimos e estudamos a “¶ controlabilidade [D(A)]-aproximada ” para o sistema de controle¶ 8<:¶ d¶ dt¶ [(I &#8722; B)x(t)] = Ax(t) + Cu(t) + f(t, x(t)), t 2 I = (0, a],¶ x(0) = x0 2 X.¶ (6)¶. Observamos que nossos comentários relacionados com a controlabilidade exata de sistemas¶ neutros, foram publicados recentemente em journal Computers & Mathematics with Applications,¶ veja [12] para detalhes. Também mencionamos que no pre-print [16] são resumidos nossos resultados¶ relacionados a existência de soluções fracas de (5) e a controlabilidade do sistema de controle (6). / In this work, we report two technical errors in some recent papers treating on existence and¶ xact controllability of solutions for a class of partial neutral functional differential control systems¶ described in the abstract form¶ 8<:¶ d¶ dt¶ (x(t) + G(t, xt)) = Ax(t) + F(t, xt) + Cu(t), t 2 I = (0, a],¶ x0 = '.¶ (1)¶ In this system A is the infinitesimal generator of an C0-semigroup of bounded linear operators¶ defined on a Banach space X; the functions xt are the histories, C : U ! X is a bounded operator,¶ U is a Banach space and and F,G are appropriate functions.¶ Additionally, by using some simple perturbation criterion, we discuss the existence of mild¶ solution for the system¶ 8<:¶ d¶ dt¶ [x(t) + Bx(t)] = Ax(t) + f(t, x(t)), t 2 I = [0, a],¶ x(0) = x0 2 X,¶ (2)¶ where B is a bounded linear operator. We also introduce the concept of approximate [D(A)]-¶ controlability and establish conditions under which the associated control system¶ 8<:¶ d¶ dt¶ [x(t) + Bx(t)] = Ax(t) + Cu(t) + f(t, x(t)), t 2 I = [0, a],¶ x(0) = x0 2 X,¶ (3)¶ is approximate [D(A)]- controllable.¶ We mention that our observation on exact controllability of abstract neutral system was recently¶ published in journal Computers & Mathematics with Applications, see [12] for details. We also note,¶ that the pre-print [16] contain some new results concerning existence of solutions and controllability¶ for the systems (2) and (3) respectively.