Algebraic Theory of Quasivarieties

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Springer Science & Business Media, Sep 30, 1998 - Mathematics - 300 pages
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The theory of quasivarieties constitutes an independent direction in algebra and mathematical logic and specializes in a fragment of first-order logic-the so-called universal Horn logic. This treatise uniformly presents the principal directions of the theory from an effective algebraic approach developed by the author himself. A revolutionary exposition, this influential text contains a number of results never before published in book form, featuring in-depth commentary for applications of quasivarieties to graphs, convex geometries, and formal languages. Key features include coverage of the Birkhoff-Mal'tsev problem on the structure of lattices of quasivarieties, helpful exercises, and an extensive list of references.
 

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Contents

Basic Notions
1
12 Constructions
11
13 Closure Operators
23
14 Congruences and Quotient Structures
32
15 Universal Horn Classes and Quasivarieties
47
Finitely Presented Structures
57
21 Defining Relations
58
22 Calculi of Atomic Formulas
70
42 Free and LowerBounded Lattices
150
43 Finite Convex Geometries
155
44 Lattices of Algebraic Subsets
159
Lattices of Quasivarieties
169
51 The Simplest Properties
170
52 Characterization of Lattices of Quasivarieties and Lattices of Varieties
183
53 Equaclosure Operators
195
54 Complete Homomorphic Images of Lattices of Quasivarieties
207

23 Characteristic Properties of Quasivarieties
77
24 Finitely Defined and LimitProjective Structures
90
25 Relative Quasivarieties and Birkhoff Classes
96
Subdirectly Irreducible Structures
103
32 Atomic Compact Structures
114
33 Residually Small Quasivarieties
125
34 Cardinalities of Subdirectly Irreducible Structures
132
Join Semidistributive Lattices
141
41 Examples and the Simplest Properties
142
55 Reduction Theorems
225
56 The BirkhoffMaltsev Problem Axiomatic Approach
238
QuasiIdentities on Structures
245
62 Infinitely Based Quasivarieties
252
63 Independent Axiomatizability and Meet Decompositions in Lattices
259
64 3Element Algebras without Independent Bases of QuasiIdentities
269
References
277
Subject Index
295
Copyright

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Page 285 - A NOTE ON THE IMPLICATIONAL CLASS GENERATED BY A CLASS OF STRUCTURES.
Page 285 - On the word problem for the modular lattice with four free generators. Math. Ann.
Page 285 - Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34-59.
Page 283 - MCKENZIE, R., Commutator Theory for Congruence Modular Varieties, London Math. Soc. Lecture Note Series, 125, Cambridge Univ. Press, Cambridge-New York, 1987.
Page 283 - Congruence lattices of congruence semidistributive algebras, Lattice Theory and its Applications (Darmstadt, 1991), 63-78, Res. Exp. Math., 23, Heldermann, Lemgo, 1995.
Page 283 - FREESE, R. and MCKENZIE, R., Residually small varieties with modular congruence lattices, Trans. Am. Math. Soc., 264, No.

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