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Proceedings of the American Mathematical Society

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

 
 

 

Two embedding theorems for lattices


Authors: G. Grätzer and C. R. Platt
Journal: Proc. Amer. Math. Soc. 69 (1978), 21-24
MSC: Primary 06A35
DOI: https://doi.org/10.1090/S0002-9939-1978-0465963-5
MathSciNet review: 0465963
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Abstract: A lattice L satisfies $ (S{D_ \wedge })$ if $ a \wedge b = a \wedge c$ implies that $ a \wedge b = a \wedge (b \vee c) ((\mathrm{SD}_\vee)$ is defined dually). Theorem. Every lattice can be embedded in the ideal lattice of a lattice satisfying $ (\mathrm{SD}_\wedge)$ (respectively, $ (\mathrm{SD}_\vee)$). Call a lattice K transferable iff whenever K can be embedded in the ideal lattice of a lattice L, then K can be embedded in L. Corollary. Every transferable lattice satisfies $ (\mathrm{SD}_\wedge)$ and $ (\mathrm{SD}_\vee)$.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0465963-5
Keywords: Lattice, semidistributive, embedding
Article copyright: © Copyright 1978 American Mathematical Society

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