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 purpose of this thesis is to investigate the idea of topological "connectedness" by presenting some of the basic ideas concerning connectedness along with several related concepts.

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

The development of the concept of a filter leads to a theory of convergence in topological spaces. There is a close relationship between the concept of a net and that of a filter.

This paper will be devoted to an exposition of some of the basic properties of paracompact spaces. In particular, it will be shown that every pseudo-metrizable space is paracompact and countably paracompact.

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.

The purpose of this paper is to define an integral for real-valued functions which are defined on a field of sets and to demonstrate several properties of such an integral.

This report presents both the most essential known results and new results in the theory of partially ordered groups and rings. This report deals with partially ordered groups and rings in an algebraic aspect because it is more important than partially ordered, fully ordered and lattice-ordered semigroup theory.

The primary purpose of this paper is to develop a rigorous study of the Schwarz derivative. This study will be based primarily on the comparison of the ordinary derivative to the Schwarz derivative.

This thesis presents a development of some useful concepts concerning topological spaces. Most of the theorems given apply to the most general form of topological space.

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.

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

This paper will be devoted to an exposition of some of the relationships existing between a field and certain of its extension fields. In particular, it will be shown that many fields may be characterized rather simply in terms of their subfields which, in turn, may be directly correlated with the subgroups of a finite group of automorphisms of the given field.

This paper is concerned with equations in which all derivatives are ordinary rather than partial derivatives. The customary meanings of differential order and difference order of an equation are observed.

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

This thesis describes semigroups and the properties of both regular and inverse semigroups.

Some of the properties of the specific summation methods will be investigated, such as what type of divergent series a method can or cannot sum, if the insertion of zeros into a series does change the sum, and when different methods give the same sum for a series.

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 a discussion on the basic fundamentals of the theory of sets. Primarily, the discussion will be confined to the study of cardinal and ordinal numbers. The concepts of sets, classes of sets, and families of sets will be undefined quantities, and the concept of the class of all sets will be avoided.

Because lattice theory is so vast, the primary purpose of this paper will be to present some of the general properties of lattices, exhibit examples of lattices, and discuss the properties of distributive and modular lattices.

This paper is primarily concerned with developing the theory of real-valued functions of bounded variation and those ideas which are closely related to this main topic. In addition to this, some emphasis has been placed on the relationship of the theory of functions of bounded variation to specific areas of analysis. In particular, integration theory has been chosen as the vehicle to demonstrate this connection.

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