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Meet-irreducible elements in implicative lattices


Author: Dorothy P. Smith
Journal: Proc. Amer. Math. Soc. 34 (1972), 57-62
MSC: Primary 06A35
DOI: https://doi.org/10.1090/S0002-9939-1972-0291035-3
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0291035-3
Keywords: Brouwerian lattice, Heyting algebra, implicative lattice, lattice, meet-irreducible, relative annihilator, relative pseudocomplement
Article copyright: © Copyright 1972 American Mathematical Society

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