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Distributivity in Lattices

Name: Distributivity in Lattices
Keywords: rings, modules, fields, 09.1X

Distributivity in lattices

descriptionPublicationkeyboard_double_arrow_right Part of book or chapter of book , Article , Other literature type 01 Dec 1952 English Publisher:Birkhäuser BostonJournal:Duke Mathematical Journal, volume 19 (issn: 0012-7094,

Authors: Dilworth, R. P.; McLaughlin, J. E.;

doi: 10.1007/978-1-4899-3558-8_22 , 10.1215/s0012-7094-52-01972-8

Distributivity in Lattices

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Abstract

A lattice L is infinitely (join) distributive if a ∩ ∪ B b = ∪ B (a ∩ b) whenever the indicated joins exist in L. Clearly infinite distributivity implies ordinary distributivity. On the other hand it is easy to give examples of distributive lattices which are not infinitely distributive. For example, the rational integers under the relation of division form a distributive lattice which is not infinitely distributive. However for complemented lattices, as observed by Tarski [8] and von Neumann [6], distributivity implies infinite distributivity.

Related Organizations

California Institute of Technology
United States

Keywords

rings, modules, fields, 09.1X

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