Meet-irreducible elements in implicative lattices

Smith, Dorothy P.

doi:10.1090/S0002-9939-1972-0291035-3

Meet-irreducible elements in implicative lattices
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by Dorothy P. Smith

Proc. Amer. Math. Soc. 34 (1972), 57-62

DOI: https://doi.org/10.1090/S0002-9939-1972-0291035-3

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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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