## Algebraic Theory of QuasivarietiesThe 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 |

277 | |

295 | |

### Common terms and phrases

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

### Popular passages

Page 285 - A NOTE ON THE IMPLICATIONAL CLASS GENERATED BY A CLASS OF STRUCTURES.

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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.