## Introduction to Lattices and OrderThis new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. |

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### Contents

III | 1 |

IV | 5 |

V | 10 |

VI | 14 |

VII | 20 |

VIII | 23 |

IX | 25 |

X | 33 |

XXXVI | 141 |

XXXVII | 145 |

XXXIX | 148 |

XL | 155 |

XLI | 165 |

XLII | 169 |

XLIII | 175 |

XLIV | 180 |

XI | 39 |

XII | 41 |

XIII | 46 |

XIV | 50 |

XV | 53 |

XVI | 56 |

XVII | 65 |

XIX | 74 |

XX | 79 |

XXI | 85 |

XXIII | 88 |

XXIV | 93 |

XXV | 96 |

XXVI | 104 |

XXVII | 112 |

XXVIII | 114 |

XXIX | 116 |

XXX | 119 |

XXXI | 124 |

XXXII | 130 |

XXXIV | 134 |

XXXV | 137 |

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### Common terms and phrases

algebraic lattice assume axioms bijective Boolean algebra Boolean lattice Boolean space Boolean term bottom element chain Chapter clopen down-sets clopen subsets closure operator compact complete lattice computer science concept lattice congruence Consider context continuous maps CPO Fixpoint Theorem defined definition denoted diagram directed set directed subset disjoint distributive lattice DM(P domain dual duality dually example Exercise exists F is order-preserving fi(F Figure filter finite distributive lattice finite lattice finite ordered sets finite subset following are equivalent Galois connection given Hence Hint homomorphism implies induction infinite information systems isomorphic join-dense join-irreducible elements joins and meets lattice of sets lattice theory least fixpoint least upper bound Lemma map F maximal element meet-dense modular non-empty subset numbers order-embedding order-isomorphism order-preserving order-preserving map partial maps pre-CPO Priestley space prime ideal proof Proposition Prove semantic set and let Show strings structure SubG subgroups sublattice topological space topology union wffs