## Lattice Theory, Volume 25, Part 2Since its original publication in 1940, this book has been revised and modernized several times, most notably in 1948 (second edition) and in 1967 (third edition). The material is organized into four main parts: general notions and concepts of lattice theory (Chapters I-V), universal algebra (Chapters VI-VII), applications of lattice theory to various areas of mathematics (Chapters VIII-XII), and mathematical structures that can be developed using lattices (Chapters XIII-XVII). At the end of the book there is a list of 166 unsolved problems in lattice theory, many of which still remain open. It is excellent reading, and ... the best place to start when one wishes to explore some portion of lattice theory or to appreciate the general flavor of the field. --Bulletin of the AMS |

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### Contents

ix | |

POSTULATES FOR LATTICES | 20 |

STRUCTURE AND REPRESENTATION THEORY | 55 |

GEOMETRIC LATTICES | 80 |

COMPLETE LATTICES | 111 |

UNIVERSAL ALGEBRA | 132 |

APPLICATIONS TO ALGEBRA | 159 |

TRANSFINITE INDUCTION | 180 |

CHAPTER XII | 277 |

LATTICEORDERED GROUPS | 287 |

LATTICEORDERED MONOIDS | 319 |

CHAPTER XV | 347 |

POSITIVE LINEAR OPERATORS | 380 |

LATTICEORDERED RINGS | 397 |

411 | |

417 | |

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### Common terms and phrases

Abelian Archimedean automorphisms Banach lattice binary Boolean algebra Boolean lattice Borel algebra Brouwerian lattice cardinal number chain condition Chapter closed sets closure commutative compact complemented lattice complemented modular lattice complete lattice completing the proof congruence relations construct contains Conversely COROLLARY countable deﬁned deﬁnition direct product directed vector space distributive lattice dual ideal dually elements epimorphic Example Exercises exists ﬁeld ﬁnite ﬁnite length ﬁrst functions geometric lattice given Hence identity implies inﬁnite integrally closed intersection interval isomorphic isotone join l-group l-group G l-groupoid l-ideals l-monoid l-ring lattice of ﬁnite LEMMA Math maximal metric lattice modular lattice Moreover morphism nonvoid open sets ordered group ordinal ortholattice permutable po-group points polynomials poset prime relatively complemented relatively complemented lattice result ring satisﬁes semimodular lattice sequence Show subalgebra subdirect product sublattice subspaces sufﬁcient theory topological space topology trivial unique upper bound valuation vector lattice