Hyperidentities are important formulae of second-order logic, and research in hyperidentities paves way for the study of second-order logic and second-order model theory.
This book illustrates many important current trends and perspectives for the field of hyperidentities and their applications, of interest to researchers in modern algebra and discrete mathematics. It covers a number of directions, including the characterizations of the Boolean algebra of n-ary Boolean functions and the distributive lattice of n-ary monotone Boolean functions; the classification of hyperidentities of the variety of lattices, the variety of distributive (modular) lattices, the variety of Boolean algebras, and the variety of De Morgan algebras; the characterization of algebras with aforementioned hyperidentities; the functional representations of finitely-generated free algebras of various varieties of lattices and bilattices via generalized Boolean functions (De Morgan functions, quasi-De Morgan functions, super-Boolean functions, super-De Morgan functions, etc); the structural results for De Morgan algebras, Boole-De Morgan algebras, super-Boolean algebras, bilattices, among others.
While problems of Boolean functions theory are well known, the present book offers alternative, more general problems, involving the concepts of De Morgan functions, quasi-De Morgan functions, super-Boolean functions, and super-De Morgan functions, etc. In contrast to other generalized Boolean functions discovered and investigated so far, these functions have clearly normal forms. This quality is of crucial importance for their applications in pure and applied mathematics, especially in discrete mathematics, quantum computation, quantum information theory, quantum logic, and the theory of quantum computers.
Contents:
- Preface
- Introduction
- Some Basic Concepts
- Hyperidentities of Lattices
- Boole-De Morgan Algebras: Quasi-De Morgan Functions
- De Morgan Algebras: De Morgan Functions
- Hyperidentities of Boolean Algebras: Super-Boolean Algebras
- Elementary Theories of Super-Boolean Algebras
- Hyperidentities of De Morgan Algebras
- Functional Completeness Theorem for De Morgan Functions
- Bilattices
- Super-Boolean Functions and Free Super-Boolean Algebras
- Super-De Morgan Functions and Free Super-De Morgan Algebras
- Bi-De Morgan Functions and Free Distributive Bilattices
- Weakly Idempotent Lattices and Bilattices
- A Set-Theoretical Representation for Weakly Idempotent Lattices and Interlaced Weakly Idempotent Bilattices
- Bigroups and Gratzer Algebras
- Abelian Algebras
- Non-trivial Associative and Distributive Hyperidentities in q-Algebras
- Essentially Non-trivial Associative and Distributive Hyperidentities in Semigroups
- Binary Representations of Semigroups: The Multiplicative Groups of Fields and Hyperidentities
- Associative Formulae with Functional Variables
- Distributive Formulae with Functional Variables
- Other Open Problems
- Bibliography
- Index
Readership: Undergraduate, graduate students, researchers.
Key Features:
- The book brings together and advances several key directions of research in hyperidentities as well as solutions to important problems, including Plotkin's problem on functional representation of free algebras and Jacobson-Maltsev-Fuchs problem on the characterization of multiplicative groups of fields
- The normal-form generalized Boolean functions discovered in the book are important for their applications in pure and applied mathematics, especially in discrete mathematics, quantum computation, quantum information theory, quantum logic and the theory of quantum computers
- The book provides the algebraic foundations for the study of the properties of algebras in the language of second-order logic
