Boolean algebras have historically played a special role in the development of the theory of general or "universal" algebraic systems, providing important links between algebra and analysis, set theory, mathematical logic, and computer science. It is not surprising then that focusing on specific properties of Boolean algebras has lead to new directions in universal algebra.
In the first unified study of polynomial completeness, Polynomial Completeness in Algebraic Systems focuses on and systematically extends another specific property of Boolean algebras: the property of affine completeness. The authors present full proof that all affine complete varieties are congruence distributive and that they are finitely generated if and only if they can be presented using only a finite number of basic operations. In addition to these important findings, the authors describe the different relationships between the properties of lattices of equivalence relations and the systems of functions compatible with them.
An introductory chapter surveys the appropriate background material, exercises in each chapter allow readers to test their understanding, and open problems offer new research possibilities. Thus Polynomial Completeness in Algebraic Systems constitutes an accessible, coherent presentation of this rich topic valuable to both researchers and graduate students in general algebraic systems.
Table of Contents
ALGEBRAS, LATTICES, AND VARIETIES
Algebras, Languages, Clones, Varieties
CHARACTERIZATIONS OF EQUIVALENCE LATTICES
Compatible Function Lifting
PRIMALITY AND GENERALIZATIONS
Primality and Functional Completeness
Near Unanimity Varieties
Generalizations of Primality
AFFINE COMPLETE VARIETIES
Introduction and Instructive Examples
Varieties with a Finite Residual Bound
Locally Finite Affine Complete Varieties
POLYNOMIAL COMPLETENESS IN SPECIAL VARIETIES
Strictly Locally Affine Complete Algebras
Algebras Based on Distributive Lattices
"This book gives a thorough, systematic treatment of various notions of polynomial completeness … the book is overdue as a reference for universal algebraists."
Mathematical Reviews, 2003a