Intervals in lattices of κ-meet-closed subsets

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  • Marcel Erné
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Original languageEnglish
Pages (from-to)137-153
Number of pages17
JournalORDER
Volume21
Issue number2
Publication statusPublished - 18 Aug 2004

Abstract

We study abstract properties of intervals in the complete lattice of all κ-meet-closed subsets (κ-subsemilattices) of a κ-(meet-) semilattice S, where κ is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closure of A ∪ {x} leaves a closed set). Such closure systems have many pleasant geometric and lattice-theoretical properties; for example, they are always weakly atomic, lower locally Boolean and lower semimodular, and each member has a decomposition into completely join-irreducible elements. For intervals of κ-subsemilattices, we describe the covering relation, the coatoms, the ∨-irreducible and the ∨-prime elements in terms of the underlying κ-semilattices. Although such intervals may fail to be lower continuous, they are strongly coatomic if and only if every element has an irredundant (and even a least) join-decomposition. We also characterize those intervals which are Boolean, distributive (equivalently: modular), or semimodular.

Keywords

    (Strongly) coatomic, (Weakly) atomic, Complete lattice, Extremally detachable, Interval, Irreducible, Meet-closed, Prime, Semilattice

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Cite this

Intervals in lattices of κ-meet-closed subsets. / Erné, Marcel.
In: ORDER, Vol. 21, No. 2, 18.08.2004, p. 137-153.

Research output: Contribution to journalArticleResearchpeer review

Erné M. Intervals in lattices of κ-meet-closed subsets. ORDER. 2004 Aug 18;21(2):137-153. doi: 10.1007/s11083-004-3716-2
Erné, Marcel. / Intervals in lattices of κ-meet-closed subsets. In: ORDER. 2004 ; Vol. 21, No. 2. pp. 137-153.
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