Let X and Y be topological spaces and C and D semigroups under composition of maps from X to X and Y to Y respectively. Let H be an isomorphism from C to D; it is shown that if both C and D contain the constant maps then there exists a bijection h from X to Y such that H(f) = h∘f∘h⁻¹, VfɛC. We investigate this situation and find sufficient conditions for this h to be a homeomorphism. In this regard we study the familiar semigroups of continuous, closed, and connected maps. An auxiliary problem is the case when C = D and H is an automorphism of D), We then ask when is every automorphism is inner. The question is answered for certain particular semigroups; e.g., the semigroup of differentiable maps on the reals has the property that all automorphisms are inner. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis consists of two parts. In the first part, we study summability in left amenable semigroups. More explicitly, various summability methods defined by matrices are considered. Necessary and (or) sufficient conditions are given for matrices to be regular, almost regular, Schur, almost Schur, strongly regular and almost strongly regular, generalizing those of O. Toeplitz, J. P. King, J. Schur, G. G. Lorentz and P. Schaefer for the semigroup of additive positive integers. The theorems are of interest even for the semigroup of multiplicative positive integers. Let S be a topological semigroup which is amenable as a discrete semigroup. Denote by LUC(S) the set of bounded real-valued left uniformly continuous functions on S. It is shown by E. Granirer that if S is a separable topological group which is amenable as a discrete group and has a certain property (B) then LUC(S) has "many" left invariant means. In the second part of this thesis, we extend this result to certain topological subsemigroups of a topological group. In particular, we show that if S is a separable closed non-compact subsemigroup of a locally compact group which is amenable as a discrete semigroup then LUC(S) has "many" left invariant means. Finally, an example is given to show that this result cannot be extended to every topological semigroup. / Science, Faculty of / Mathematics, Department of / Graduate

No description available.

Baseado no conjunto das funções parciais injetoras em um conjunto não vazio e no conjunto das funções booleanas de várias variáveis, a dissertação apresenta os conceitos de grupos e de semigrupos inversos, que são constituídos por elementos inversíveis. No caso de grupos, a definição de elemento inversível é a usual e, no caso de semigrupos inversos, a definição de elemento inversível é uma generalização do conceito usual de elemento inversível. / Based on the set of injective partial functions in a non empty set and in the set of booleans functions with many variables, this paper shows the concepts of groups and inverse semigroups, which are both made of inversible elements. In groups, the definition of inversible element is the usual and, in inverse semigroups, the definition of inversible element is a generalization of the usual concept of inversible element.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

W. D. Munn proved that a finite dimensional representation of an inverse semigroup is equivalent to a ⋆-representation if and only if it is bounded. The first goal of this thesis will be to give new analytic proof that every finite dimensional representation of a compact inverse semigroup is equivalent to a ⋆-representation. The second goal is to parameterize all finite dimensional irreducible representations of a compact inverse semigroup in terms of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new and simpler proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Our last theorem is the following: every norm continuous irreducible ∗-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.

In this thesis we are concerned with arithmetic in a certain partially ordered, commutative semigroup D. The first chapter investigates the class of countable Boolean algebras from which this semigroup arises. The elements of D correspond to the isomorphism classes of the Boolean algebras under consideration. In Chapter 2 we begin the study of the semigroup structure of D. D is axiomatically described by three groups of axioms. It is proved that these axioms are categorical. The ordering of D is used to investigate the multiplication. The set of T of torsion elements of D (elements with only finite many distinct powers), form a subsemigroup whose structure is studied. There is a natural torsion free quotient D/T whose structure is also investigated. In Chapter 3, the axioms are used to characterize elements s of T in terms of the arithmetic in the subsemigroup generated by the elements that are smaller than s. The characterization is used to determine elements of T that cover a single element. In the last part of Chapter 3, we obtain some sufficient, purely combinatorial conditions for an element to have infinite order.

by Law Chun Kong. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliography: leaves 129-133.