Polynomial Completeness in Algebraic Systems

by Kaarli; Kalle

• ISBN13: 9781584882039
• ISBN10: 1584882034
• Format: Hardcover
• Copyright: 2000-07-21
• Publisher: Chapman & Hall/

Summary

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.

Author Biography

Kalle Kaarli is a Professor of Mathematics at the University of Tartu in Estonia.

Table of Contents

Algebras, Lattices, and Varieties	1	(64)
Algebras, languages, clones, varieties	1	(26)
Algebras	1	(4)
Languages and structures	5	(8)
Clones and related topics	13	(10)
Varieties	23	(4)
Congruence properties	27	(38)
Mal'cev conditions	27	(9)
Definable principal congruences	36	(8)
Near unanimity operations and congruence distributivity	44	(2)
Congruence permutability and rectangular subalgebras	46	(3)
Congruence distributivity and skew congruences	49	(9)
The commutator	58	(7)
Characterizations of Equivalence Lattices	65	(42)
Introduction	65	(2)
Arithmeticity	67	(21)
The strong Chinese remainder condition for equivalence lattices	68	(1)
The compatible function extension property	69	(2)
Existence of compatible Pixley functions	71	(7)
Applications of principal Pixley functions	78	(8)
Compatible choice functions	86	(2)
Compatible function lifting	88	(19)
Compatible function lifting implies distributivity (finite case)	91	(3)
Compatible function lifting implies compatible function system extension	94	(7)
Compatible function lifting implies distributivity (residually finite case)	101	(6)
Primality and Generalizations	107	(66)
Primality and functional completeness	107	(21)
Primality	107	(2)
History: Post algebras	109	(2)
Primality and functional completeness: characterizations	111	(9)
Functionally complete algebras	120	(8)
Near unanimity varieties	128	(9)
Characterizations	128	(5)
More properties of near unanimity varieties	133	(4)
Arithmetical varieties	137	(4)
Generalizations of primality	141	(24)
Generalizations involving term function characterizations	141	(16)
Affine complete algebras	157	(4)
Polynomial interpolation	161	(4)
Categorical equivalence	165	(8)
Affine Complete Varieties	173	(48)
Introduction and instructive examples	173	(4)
General properties	177	(5)
Varieties with a finite residual bound	182	(11)
Some categorical properties	182	(2)
Property (S)	184	(4)
Two counter-examples	188	(5)
Locally finite affine complete varieties	193	(28)
Existence of a near unanimity term	193	(2)
Structure theorems	195	(6)
The arithmetical case	201	(9)
Examples	210	(8)
Categorical equivalence	218	(3)
Polynomial Completeness in Special Varieties	221	(112)
Strictly locally affine complete algebras	221	(11)
Interpolation and the class SPf(A)	221	(5)
Neutrality	226	(6)
Modules	232	(22)
General observations	232	(3)
Completeness of weak direct powers	235	(5)
Semisimple modules and semisimple rings	240	(4)
Abelian groups	244	(10)
Lattices	254	(31)
General observations	254	(3)
Distributive lattices	257	(9)
Order affine completeness of lattices	266	(14)
Order functionally complete lattices	280	(5)
Algebras based on distributive lattices	285	(21)
Lattices affine complete in a distributive lattice	286	(3)
Stone algebras	289	(7)
Kleene algebras	296	(10)
Semilattices	306	(11)
Compatible functions on semilattices	306	(8)
Completeness theorems	314	(3)
Miscellaneous results	317	(16)
p-rings and subalgebra primal algebras	317	(5)
Median algebras	322	(4)
Commutative inverse semigroups	326	(4)
Lattice ordered groups	330	(3)
Bibliography	333	(18)
Index	351