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, Volume 21, Issue 2, pp 137–153 | Cite as

Intervals in Lattices of κ-Meet-Closed Subsets

  • Marcel Erné
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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

(weakly) atomic (strongly) coatomic complete lattice extremally detachable interval irreducible meet-closed prime semilattice 

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of HannoverGermany

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