## Algebraic theory of lattices |

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### Contents

Partially ordered sets | 1 |

Lattices | 8 |

Distributive modular and semimodular lattices | 19 |

Copyright | |

13 other sections not shown

### Common terms and phrases

Arguesian ascending chain condition assume assumption Boolean algebra compact elements compactly generated lattice complemented lattice complemented modular lattice complete lattice completing the proof congruence relation Consequently contains define direct product distributive lattice dual lattice element a e embedded equivalence classes exists finite dimensional finite subset follows free lattice geometric lattice geomodular lattice greatest element hence holds homomorphism implies inasmuch indecomposable induction infer infinite integer irreducible element isomorphic join of atoms lattice identity lattice of dimension lattice satisfying least element lower transpose maximal element Mobius function Moreover nonempty normalized dimension function one-to-one partially ordered set prime quotient quotient sublattice relatively complemented satisfies the ascending semimodular lattice set-union strongly atomic lattice subdirectly irreducible subquotient subspaces superdivisor Suppose theorem upper continuous upper transpose vector space weakly atomic weakly projective whence yields Zorn's lemma