## 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 |

References | 277 |

295 | |

### Common terms and phrases

algebraic algebraic lattice arbitrary assume assumption atomic atomic formulas basis of quasi-identities belongs Boolean called cardinality characterization closed closure complete conclude condition congruence Consequently consider contains continuous contradicts Conversely core Corollary decomposition defined definition denote direct product distributive dually easy edges element embedding equality equivalent example exists finite finitely based geometry given Gorbunov graph groups Hence holds homomorphism identities implies inclusion independent infinite interpretation introduce irreducible isomorphism join semidistributive L-structures language least Lemma locally finite Logic lower Lq(K mapping Math maximal means meet moreover nontrivial obtain obvious operator particular presented prevariety projective Proof Proposition prove quasi-identities quasivariety relation relation symbol relative remains respect retract rule satisfies sentence signature similar space structure subclass subdirect product subdirectly subset substructure Theorem theory variables variety