Boolean algebras and equations have been the subject of considerable theoretical and application-oriented investigation. One of the seminal works in this field, including Boolean functions and equations, has been Boolean functions and equations [1], also by Rudeanu, who has been investigating lattice theory and universal and Boolean algebra for several decades.

This book, as stated in the preface, is in many ways an update to, but also a significant expansion on Boolean functions and equations, primarily in that it considers functions and equations over general lattices, but also in the applications covered.

Chapter 1 presents a brief exposition of general equations, and equations over finite sets, particularly reproductive solutions as introduced by Löwenheim [2]. Chapters 2 and 3 concisely review the fundamental prerequisites of universal algebra and lattice theory, while chapter 4 introduces the concept of equational compactness and monocompactness, as defined by Mycielsk [3], as well as extensions for lattices and Boolean algebras.

Chapter 5 extensively covers Post algebras and functions, including embeddings in other algebras, such as Heyting algebras. This chapter also covers, in extension to Boolean functions and equations, equations over Post functions, and briefly considers matrix Post equations and inequalities.

After these preliminaries, the author considers generalized structures for Boolean functions and equations in chapter 6, introducing generalized minterms and interpolating systems, as well as an axiomatic theory of prime implicants for Boolean functions. A highly interesting line of inquiry in this chapter is devoted to the interrelation of the theory of prime implicants and resolution calculi. Chapter 6 is concluded by an exposition on generalized systems of Boolean equations.

Chapter 7 is devoted to closure operators over Boolean functions; after a brief review of general closure operators, the chapter introduces isotone, antitone, monotone, independent, and decomposition closures, as well as their applications. This is continued in chapter 8 by a discussion of functional dependencies in Boolean functions by way of a general Boolean transformation, and the characteristics of this transformation. Chapter 9 reviews both historical and current techniques for solving Boolean equations and truth equations, with a special section devoted to techniques for solving quadratic Boolean equations. This chapter also covers the literature on representations of Boolean equations in computer systems.

Chapter 10 covers parts of the field of Boolean differential calculus [4], using the axiomatic approach of Kühnrich; multi-valued logic is not covered, which limits the immediate applications of the approach described, since areas such as circuit synthesis will generally require multi-valued logic. However, in the framework of this book, such an extension would clearly have been inappropriate. Instead, supplemental material should be consulted [5]. Chapter 11 addresses decomposition methods for Boolean functions, also of particular interest in the application area of switching theory, and reviews the classical approaches to this problem. As before, multi-valued logic is not covered in this chapter, although chapter 13 contains some additional results in this area [6].

Chapter 12 investigates a number of interrelations and analogs of Boolean algebras to other mathematical areas, particularly to the Lindenbaum-Tarski algebra, and of geometric interpretations. This chapter only briefly touches on areas that are certain to elicit further interest in the future. Further applications and interrelations are covered in the final chapter, chapter 14, where, aside from the obvious applications of circuit synthesis and automata theory, areas such as graph theory and database theory are covered.

This eminently readable book is rounded out by three appendices, one of which is an errata list for Rudeanu’s other book [1], and an extensive bibliography. Just as Boolean functions and equations was, this book will be considered the main work and reference on Boolean algebra for a considerable amount of time.