In this paper, elementary properties of relative extreme points are investigated. The properties are defined in linear and topological terms. Proofs of many of these properties require the use of topological concepts.

Explores topological spaces, filters, and nets with definitions and examples.

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.

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.

The primary purpose of this thesis is to carefully develop and prove some of the fundamental, classical theorems of the differential calculus for functions of two real variables.

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 analyzes tensor associations with applications to Reimann spaces.

The purpose of this paper is to study convergence of certain continued fractions.

The purpose of this paper is to define and investigate some of the properties of chains. Particular attention is given to a natural topology for chains, called the interval topology, and how the chain properties and topological properties of chains affect each other.

This thesis provides a development of integration from two different points of view. In Chapter I, a measure and a measurable function are defined. A theory of integration is then developed in Chapter II based on the measure. In Chapter III, the integral is introduced directly without first going through the process of defining a measure, and a measure is developed from the integral. The concluding chapter shows the equivalence of the two integrals under rather general conditions.

This paper is an introductory, algebraic study of semigroups.

The objective here is to give a logical development of some mathematical systems beginning with those having relatively little algebraic structure, and progressing to systems rich in structure. The discussion will commence with a mathematical system called the groupoid, a nonempty set with a binary operation.

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 thesis discusses elliptic geometry including the order and incidence properties, projective properties and congruence properties.