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

Axioms for a topological space are generally based on neighborhoods where "neighborhood" is an undefined term. Then, limit points are defined in terms of neighborhoods. However, limit points seem to be the basic concept of a topological space, rather than neighborhoods. For this reason, it will be attempted to state a set of axioms for a topological space, using limit point as the undefined concept, and to delete the idea of neighborhoods from the theory.

One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers.

The study of metric spaces is closely related to the study of topology in that the study of metric spaces concerns itself, also, with sets of points and with a limit point concept based on a function which gives a "distance" between two points. In some topological spaces it is possible to define a distance function between points in such a way that a limit point of a set in the topological sense is also a limit point of the same set in a metric sense. In such a case the topological space is "metrizable". The real numbers with its usual topology is an example of a topological space which is metrizable, the distance function being the absolute value of the difference of two real numbers. Chapters II and III of this thesis attempt to classify, to a certain extent, what type of topological space is metrizable. Chapters IV and V deal with several properties of metric spaces and certain functions of metric spaces, respectively.

The purpose of this chapter is to develop a form of a "free" Boolean algebra with Σ as a base, by imposing the usual Boolean operations on the set Σ and thus generating new elements freely within explicitly prescribed restrictions.

The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

This paper presents an introduction to the theory of ideals in a ring with emphasis on ideals in a commutative ring with identity.

This thesis discusses groups, modules, the module of homomorphisms, and extension of modules.

Gives the basic definitions and theorems of similar partially ordered sets; studies finite partially ordered sets, including the problem of combinatorial analysis; and includes the ideas of complete, dense, and continuous partially ordered sets, including proofs.

The purpose of this paper is to present two proofs of Helly's Theorem and to use it in the proofs of several theorems classified in a group called Helly-type theorems.

This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper.

The purpose of this paper is to investigate some of the more basic properties of peripherally continuous functions, graph maps and connectivity maps.

A set of definitions, theorems and proofs to describe the Laplace transformation.

This thesis investigates some of the properties of certain transformations. Some properties are considered in general; others, only in the xy-plane.

The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.

This thesis explores the fundamental properties of the contingent.

The purpose of this paper will be to investigate certain properties of algebraic systems known as rings.

This study of the Euclidean N-space looks at some definitions and their characteristics, some comparisons, boundedness and compactness, and transformations and mappings.

It may be said of certain pairs of elements of a set that one element precedes the other. If the collection of all such pairs of elements in a given set exhibits certain properties, the set and the collection of pairs is said to constitute a partially ordered set. The purpose of this paper is to explore some of the properties of partially ordered sets.

This paper consists of a study of the direct sum U of two rings S and T. Such a direct sum is defined as the set of all ordered pairs (s1, t1), where s1 is an arbitrary element in S and t1 is an arbitrary element in T.

This thesis describes properties of Abelian groups, and develops a study of the properties of divisibility in Abelian groups.

The purpose of this paper is to present some fundamental properties of algebraic semigroups. The development of the theory of semigroups has appeared for the most part in the past few years of this century. A semigroup is the result of a weakening of the axioms for a group. Thus all groups are semigroups. That the study of semigroups is very closely related to the abstract study of general transformations is, perhaps, one of the reasons for the rapid development of semigroup theory.

The purpose of this paper is to study two differential equations. A method of approximation by iteration is used to define sequences of functions which converge to solutions of these equations. Some properties of the solutions are proved for general boundary conditions and certain special solutions are studied in detail.