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E-raamat: Algebras, Lattices, Varieties

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This book is the third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.
List of Figures
ix
Preface for Volumes II and III xi
Acknowledgments xvii
Chapter 9 Finite Algebras and Their Clones
1(124)
9.1 Clones
1(6)
9.2 Polarity in Clone Theory
7(19)
9.3 The Spectrum of the Polar of a Clone
26(10)
9.4 The Lattice of Clones on a 2-Element Set
36(13)
9.5 Primal and Quasiprimal Algebras
49(13)
9.6 The Maximal Clones
62(25)
9.7 Properties of "Almost All" Finite Algebras
87(19)
9.8 TC Clones
106(10)
9.9 Free Spectra
116(9)
Chapter 10 Abstract Clone Theory
125(146)
10.1 Abstract Clones and their Homomorphisms
127(15)
10.2 Representations of Clones
142(16)
10.3 Presentations of Clones
158(9)
10.4 Clone Homomorphisms and Concrete Functors
167(16)
10.5 Products and CoProducts of Clones
183(16)
10.6 [ K]th Power Varieties
199(17)
10.7 Category Equivalence of Varieties
216(16)
10.8 The Interpretability Lattice--Basic Facts
232(8)
10.9 The Interpretability Lattice--Filters and Mal'tsev Conditions
240(20)
10.10 Hypervarieties
260(11)
Chapter 11 Commutator Theory
271(136)
11.1 Introduction
271(1)
11.2 Centrality Relations, Term Conditions and Commutators
272(7)
11.3 Examples
279(4)
11.4 Congruence Modular Varieties
283(6)
11.5 Extensions to Non-Modular Varieties
289(42)
11.6 Solvability, Nilpotence, the Ring and Abelian Varieties
331(14)
11.7 Residual Smallness
345(10)
11.8 Finite Axiomatizability
355(29)
11.9 Locally Finite Varieties with Restrictions on Finite Algebras
384(23)
Bibliography 407(16)
Index 423
Ralph S. Freese, University of Hawaii, Honolulu, HI.

Ralph N. McKenzie, Vanderbilt Univesity, Nashville, TN.

George F. McNulty, University of South Carolina, Columbia, SC.

Walter F. Taylor, University of Colorado, Boulder, CO.