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.

This paper establishes a basis for abelian categories, then gives the statement and proof of two equivalent definitions of an abelian category, the development of the basic theory of such categories, and the proof of some theorems involving this basic theory.

In mathematical physics, Laplace's equation plays an especially significant role. It is fundamental to the solution of problems in electrostatics, thermodynamics, potential theory and other branches of mathematical physics. It is for this reason that this investigation concerns the development of some general properties of continuous solutions of this equation.

The purpose of this thesis is to investigate certain properties of rings, ideals, and a special type of ring called a Boolean ring.

A special case of the linear programming problem, the transportation problem, is the subject of this thesis. The development of a solution to the transportation problem is based on fundamental concepts from the theory of linear algebra and matrices.

In this thesis we consider integrals of a certain class of interval functions. Specifically we consider a nondegenerate number interval [a,b], a real valued function m, defined and nondecreasing on [a,b], and the set Hm, of real valued functions f, defined on [a,b] such that: 1) f(a)=0; 2) for each subinterval [p,q] of [a,b], if m(q)-m(p)=0, then f(q)-f(p)=0; and 3) the set of all sums of the form Σ(Δf)2/Δm for subdivisions D of [a,b] is bounded above.

The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface.

This thesis is an investigation of some of the properties of polynomial rings, unique factorization domains, Euclidean domains, and principal ideal domains. The nature of some of the relationships between each of the above systems is also developed in this paper.

This thesis describes various forms of metric spaces and establishes some of the properties of functions defined on metric spaces. No attempt is made in this paper to examine a particular type of function in detail. Instead, some of properties of several kinds of functions will be observed as the functions are defined on various forms of metric spaces such as connected spaces, compact spaces, complete spaces, etc.