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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Meet-irreducible elements in implicative lattices

Author: Dorothy P. Smith
Journal: Proc. Amer. Math. Soc. 34 (1972), 57-62
MSC: Primary 06A35
MathSciNet review: 0291035
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Abstract: A characterization of meet-irreducible elements and atoms in an implicative lattice is obtained and used to derive the following theorems. A complete lattice is implicative and every element has a meet-irreducible decomposition if and only if there are enough principal prime relative annihilator ideals to separate distinct elements. The MacNeille completion of an implicative lattice is an implicative lattice; furthermore the embedding preserves relative pseudocomplements, meet-irreducible elements and atoms.

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Keywords: Brouwerian lattice, Heyting algebra, implicative lattice, lattice, meet-irreducible, relative annihilator, relative pseudocomplement
Article copyright: © Copyright 1972 American Mathematical Society