Function Algebras on Finite Sets Basic Course on Many-Valued Logic and Clone Theory

Functions which are defined on finite sets occur in almost all fields of mathematics. For more than 80 years algebras whose universes are such functions (so-called function algebras), have been intensively studied.

This book gives a broad introduction to the theory of function algebras and leads to the cutting edge of research. To familiarize the reader from the very beginning on with the algebraic side of function algebras the more general concepts of the Universal Algebra is given in the first part of the book. The second part on fuction algebras covers the following topics: Galois-connection between function algebras and relation algebras, completeness criterions, clone theory.

This book is an insdispensible source on function algebras for graduate students and researchers in mathematical logic and theoretical computer science.

Preliminaries.- Preliminaries.- Universal Algebra.- Basic Concepts of Universal Algebra.- Lattices.- Hull Systems and Closure Operators.- Homomorphisms, Congruences, and Galois Connections.- Direct and Subdirect Products.- Varieties, Equational Classes, and Free Algebras.- Function Algebras.- Basic Concepts, Notations, and First Properties.- The Galois-Connection Between Function- and Relation-Algebras.- The Subclasses of P2.- The Subclasses of Pk Which Contain Pk1.- The Maximal Classes of Pk.- Rosenberg’s Completeness Criterion for Pk.- Further Completeness Criteria.- Some Properties of the Lattice .- Congruences and Automorphisms on Function Algebras.- The Relation Degree and the Dimension of Subclasses of Pk.- On Generating Systems and Orders of the Subclasses of Pk.- Subclasses of Pk,2.- Classes of Linear Functions.- Submaximal Classes of P3.- Finite and Countably Infinite Sublattices of Depth 1 or 2 of .- The Maximal Classes of ?a?Q Polka for Q Ek.- Maximal Classes of PolkEl for 2 ? l < k.- Further Submaximal Classes of Pk.- Minimal Classes and Minimal Clones of Pk.- Partial Function Algebras.