In this thesis we will be exclusively considering uncountable groups and semigroups. Roughly speaking the underlying problem is to find “large” subgroups (or subsemigroups) of the object in question, where we consider three different notions of “largeness”: we classify all the subsemigroups of the set of all mapping from a countable set back to itself which contains a specific uncountable subsemigroup; we investigate topological “largeness”, in particular subgroups which are finitely generated and dense; we investigate if it is possible to find an integer r such that any countable collection of elements belongs to some r-generated subsemigroup, and more precisely can these elements be obtained by multiplying the generators in a prescribed fashion.

We consider a natural extension to the definition of M-automata which allows the automaton to make use of more of the structure of the monoid M, and by removing the reliance on an identity element, allows the definition of S-automata for S an arbitrary semigroup. In the case of monoids, the resulting automata are equivalent to valence automata with rational target sets which arise in the theory of regulated rewriting. We focus on the polycyclic monoids, and show that for polycyclic monoids of rank 2 or more they accept precisely the context-free languages. The case of the bicyclic monoid is also considered. In the process we prove a number of interesting results about rational subsets in polycyclic monoids; as a consequence we prove the decidability of the rational subset membership problem, and the closure of the class of rational subsets under intersection and complement. In the case of semigroups, we consider the important class of completely simple and completely 0-simple semigroups, obtaining a complete characterisation of the classes of languages corresponding to such semigroups, in terms of their maximal subgroups. In the process, we obtain a number of interesting results about rational subsets of Rees matrix semigroups.

<p> Commutative semigroup equations are described, and rules of inference for them are given. Then a skeleton sublattice of the lattice of equational classes of commutative semigroups is described, and a partial description is given of the way in which the rest of the lattice hangs on the skeleton.</p> / Thesis / Doctor of Philosophy (PhD)

No description available.

R. J. Warne has defined an wⁿ-right cancellative semigroup to be a right cancellative semigroup with identity whose ideal structure is order isomorphic to (I^o)ⁿ, where I^o is the set of non-negative integers and n is a natural number, under the reverse lexicographic order. Warne has described, modulo groups, the structure of such semigroups ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811]. He has used this structure and the theory of right cancellative semigroups having identity on which Green's relation J: is a congruence to describe the homomorphisms of an ωⁿ-right cancellative semigroup into an ωⁿ-right cancellative semigroup when 1 ≤ n ≤ 2 and m ≤ n ["Lectures in Semigroups," West Virginia Univ., unpublished]. We have described, modulo groups, the homomorphisms of an ωⁿ-right cancellative semigroup into an ω^m-right cancellative semigroup for arbitrary natural numbers n and m. One of the main results is the following: Theorem: Let P = (G ,(I^o)ⁿ , γ₁,...,γ_n, w₁,…,w_Ø(n)) and P^* = (G ,(I^o)ⁿ , α₁,...,α_n, t₁,…,t_Ø(n)) be ωⁿ-right cancellative semigroups where Ø(x) = ½x(x-1). Let z₁, ... ,z_n be elements of G^* and let f be a homomorphism of G into G^* such that (1) (Af)^(U_kg)C_{z_k} = (Aγ_kf) for A ∈ G where 1 ≤ k ≤ n and (2) ((z_k+s)^(U_kg)(U_kg)^(U_k+sg)C_{z_k} = w_Ø(n-k)+sf where 1 ≤ k ≤ n and 1 ≤ s ≤ n - k. The elements U_k (1 ≤ k ≤ n) are generators of (I^o)ⁿ, xC_{z_k} = z_kxz_k⁻¹ for x ∈ G^*, and x^a,a^b in G^* (x ∈ G^*; a,b ∈ (I^o)ⁿ are specified. Define, for (A,a₁,...,a_n) ∈ P, (A,a₁,...,a_n)M = [(Af)(a₁,...,a_n)h,(a₁,...,a_n)g] where h is a specified function from (I^o)ⁿ into G* and g is a determined endomorphism of (I^o)ⁿ. Then, M is a homomorphism of P into P* and every homomorphism of P into P* is obtained in this fashion. M is an isomorphism if and only if f and g are isomorphisms. M is onto when g is the identity and f is onto. Results similar to this theorem have been obtained when P* is an ω^m-right cancellative semigroup with m < n and m > n. Let I be the set of integers. Let S be a bisimple semigroup and let E_S denote the set of idempotents of S. S is called ωⁿ-bisimple if and only if E_S, under its natural order, is order isomorphic to I x (I^o)ⁿ under the reverse lexicographic order n ≥ 1. S is called I-bisimple if and only if E_S, under its natural order, is order isomorphic to I under the reverse usual order. Warne has described, modulo groups, the structure of ωⁿ-bisimple, ωⁿI-bisirnple and I-bisimple semigroups in ["Bisimple Inverse Semigroups Mod Groups," Duke Math. J., Vol. 14 (1967), pp. 787-811], ["ωⁿI-bisimple Semigroups," to appear], and ["I-bisimple Semigroups," Trans. Amer. Math. Soc., Vol. 130 (1968), pp. 367-386] respectively. We have described the homomorphisms of S into S* , by use of the homomorphism theory of ωⁿ-right cancellative semigroups, for the cases (i) S ωⁿ-bisimple and S* ω^m-bisimple and (ii) S I-bisimple or ωⁿI-bisimple and S* I-bisimple or ω^mI-bisimple where m and n are natural numbers. The homomorphisms of S onto S* are specified for cases (i) and (ii). Warne has determined the homomorphisms of S onto S* in certain of these cases as he studied the extensions and the congruences of ωⁿ-bisimple, ωⁿI-bisimple, and I-bisimple semigroups. Papers on these subjects are to appear at some later date. / Ph. D.

O objetivo principal deste trabalho é o estudo da estabilidade estrutural dos atratores de semigrupos. Começamos este trabalho apresentando o conceito e propriedades básicas de semigrupos que possuem atratores globais. Estudamos, então, semigrupos gradientes e dinamicamente gradientes, mostrando que eles são equivalentes e que uma pequena perturbação autônoma de um semigrupo gradiente continua sendo gradiente. Estudamos as variedades estável e instável de um ponto de equilíbrio hiperbólico e o comportamento de soluções periódicas sob perturbação. Concluímos este trabalho com o estudo dos semigrupos Morse-Smale. / The main goal of this work is the study of structural stability of global attractors. We start this work by presenting the concept and basic properties of semigroups and global attractors. We then studied gradient and dinamically gradient semigroups, showing that these concepts are equivalent and that a small autonomous pertubation of a gradient semigroup remains a gradient semigroup. We studied the stable and unstable manifolds in the neighbourhood of a hyperbolic equilibrium point and the behavior of periodic solutions under perturbation. Finally, we studied the Morse-Smale semigroups.

O objetivo principal deste trabalho é o estudo da estabilidade estrutural dos atratores de semigrupos. Começamos este trabalho apresentando o conceito e propriedades básicas de semigrupos que possuem atratores globais. Estudamos, então, semigrupos gradientes e dinamicamente gradientes, mostrando que eles são equivalentes e que uma pequena perturbação autônoma de um semigrupo gradiente continua sendo gradiente. Estudamos as variedades estável e instável de um ponto de equilíbrio hiperbólico e o comportamento de soluções periódicas sob perturbação. Concluímos este trabalho com o estudo dos semigrupos Morse-Smale. / The main goal of this work is the study of structural stability of global attractors. We start this work by presenting the concept and basic properties of semigroups and global attractors. We then studied gradient and dinamically gradient semigroups, showing that these concepts are equivalent and that a small autonomous pertubation of a gradient semigroup remains a gradient semigroup. We studied the stable and unstable manifolds in the neighbourhood of a hyperbolic equilibrium point and the behavior of periodic solutions under perturbation. Finally, we studied the Morse-Smale semigroups.

We consider naturally occurring, uncountable transformation semigroups S and investigate the following three questions. (i) Is every countable subset F of S also a subset of a finitely generated subsemigroup of S? If so, what is the least number n such that for every countable subset F of S there exist n elements of S that generate a subsemigroup of S containing F as a subset. (ii) Given a subset U of S, what is the least cardinality of a subset A of S such that the union of A and U is a generating set for S? (iii) Define a preorder relation ≤ on the subsets of S as follows. For subsets V and W of S write V ≤ W if there exists a countable subset C of S such that V is contained in the semigroup generated by the union of W and C. Given a subset U of S, where does U lie in the preorder ≤ on subsets of S? Semigroups S for which we answer question (i) include: the semigroups of the injec- tive functions and the surjective functions on a countably infinite set; the semigroups of the increasing functions, the Lebesgue measurable functions, and the differentiable functions on the closed unit interval [0, 1]; and the endomorphism semigroup of the random graph. We investigate questions (ii) and (iii) in the case where S is the semigroup Ω[superscript Ω] of all functions on a countably infinite set Ω. Subsets U of Ω[superscript Ω] under consideration are semigroups of Lipschitz functions on Ω with respect to discrete metrics on Ω and semigroups of endomorphisms of binary relations on Ω such as graphs or preorders.

The purpose of this thesis is to explore some of the characteristics of 0-simple semigroups and completely 0-simple semigroups.

The qualitative behavior of solutions of differential equations mainly addresses the various questions arising in the study of the long run behavior of solutions. The contents of this thesis are related to three of the major problems of the qualitative theory, namely the stability, the boundedness and the periodicity of the solution. Learning the qualitative behavior of such solutions is crucial part of the theory of differential equations. It is important to know if a solution is bounded or unbounded or if a solution is stable. Moreover, the periodicity of a solution is also of great significance for practical purposes.