Algebraic Theory of Quasivarieties
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.
What people are saying - Write a review
We haven't found any reviews in the usual places.
13 Closure Operators
14 Congruences and Quotient Structures
15 Universal Horn Classes and Quasivarieties
Finitely Presented Structures
21 Defining Relations
22 Calculi of Atomic Formulas
42 Free and LowerBounded Lattices
43 Finite Convex Geometries
44 Lattices of Algebraic Subsets
Lattices of Quasivarieties
51 The Simplest Properties
52 Characterization of Lattices of Quasivarieties and Lattices of Varieties
53 Equaclosure Operators
54 Complete Homomorphic Images of Lattices of Quasivarieties
23 Characteristic Properties of Quasivarieties
24 Finitely Defined and LimitProjective Structures
25 Relative Quasivarieties and Birkhoff Classes
Subdirectly Irreducible Structures
32 Atomic Compact Structures
33 Residually Small Quasivarieties
34 Cardinalities of Subdirectly Irreducible Structures
Join Semidistributive Lattices
41 Examples and the Simplest Properties
55 Reduction Theorems
56 The BirkhoffMaltsev Problem Axiomatic Approach
QuasiIdentities on Structures
62 Infinitely Based Quasivarieties
63 Independent Axiomatizability and Meet Decompositions in Lattices
64 3Element Algebras without Independent Bases of QuasiIdentities
algebraic lattice algebraic subset arbitrary assume atomic formulas axiomatizable axiomatizable class basis of quasi-identities Birkhoff Boolean cardinality closure operator color-family compact element complete lattice congruence Consequently consider contains contradicts convex geometry Corollary decomposition defined definition denote direct product distributive lattice dually Dziobiak easy embedding equaclosure operator equivalent exists a homomorphism finite lattice finite number finite signature finite structures finitely based finitely presented following assertion free lattice Gorbunov graph H Hence homomorphic image homomorphism f identities implies infinite irreducible structures isomorphism join semidistributive lattice K-congruence L-structures lattices of quasivarieties least element Lemma locally finite locally finite quasivariety lower-bounded lattice Lq(K Mal'tsev maximal modular lattices moreover nonempty nontrivial obtain obvious partially ordered set prevariety Proof Proposition prove Q-lattices quasi-Birkhoff quasivariety relation symbol retract satisfies semilattice sentence struc subclass subdirect product subdirectly irreducible sublattice subquasivariety subsemilattice substructure theory ultraproducts universal Horn class variety
Page 285 - A NOTE ON THE IMPLICATIONAL CLASS GENERATED BY A CLASS OF STRUCTURES.
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.