On the theory of Boolean vector spaces
Authors:   , Stroup Fred Oliver Jr. 
Published: February 2025

"The concept of a (abstract) Boolean vector space is due to Subrahmanyam [6]1. In introducing this concept, Subrahmanyam was motivated by Foster [1] who demonstrated that each element of a p-ring R (with unity) can be represented as a type of "Boolean vector" over the Boolean algebra of all the idempotent elements of R. In this representation, Foster made use of a "basis" of R consisting of the nonzero elements of the additive subgroup of R generated by its unity element. Shorter proofs of this latter result concerning a "basis" of R have been given by Penning [3] and Zemmer [9]. The notion of a (abstract) Boolean vector space is a rather natural generalization of this idea of considering a p-ring as a kind of "Boolean vector space" over its Boolean algebra of Idempotent elements. The development of the theory of Boolean vector spaces was continued in subsequent publications by Subrahmanyam [7], [8] and by Jagannadham [2]. Since the postulates for a Boolean vector space bear a strong resemblence to those of a vector space over a field, the study of Boolean vector spaces is influenced to a considerable extent by a comparison of the two mathematical systems. There are many very natural questions concerning the structure of Boolean vector spaces and their associated linear homomorphism (transformation) spaces which at present remain unanswered. The purpose of this dissertation will be to continue the development of the theory of Boolean vector spaces while attempting to provide answers to some of these questions. Chapter II is devoted to summarizing the definitions and previously published results concerning Boolean vector spaces which are deemed essential in the development of subsequent chapters. In Chapter III, the concept of a direct sum of Boolean vector spaces is introduced. Such direct sums play an important role in the structure theory of linear homomorphism spaces of Boolean vector spaces. Subspaces and quotient spaces of Boolean vector spaces are defined and studied in Chapter IV. In [2], Jagannadham continued the study of linear homomorphisms (transformations) of a Boolean vector space into itself which was initiated by Subrahmanyam [7]. In Chapter V, the notion of a linear homomorphism is generalized to include mappings of one Boolean vector space into another Boolean vector space. In addition to generalizing several of the results appearing in [2] and [7], some interesting results are obtained concerning direct sum representations of linear homomorphism spaces. As in the study of vector spaces over a field, the idea of a linear functional arises as a special type of linear homomorphism. Chapter VI is concerned with the theory of linear functionals. The concluding Chapter VII introduces the concept of a duality relationship among Boolean vector spaces. The principal theorem of Chapter VII provides necessary and sufficient conditions for two Boolean vector spaces to be dual spaces. Within each chapter, definitions, lemmas, and theorems are assigned numbers according to the order of their appearance within the chapter. Also, certain key statements throughout each chapter are assigned numbers. A typical reference such as (2.4) will refer to statement (2.4) which will occur as the fourth numbered statement in Chapter II. Theorems, lemmas, and definitions will be referred to as such together with their appropriate numbers."--Introduction.