Paper Info
Title
Intervals in Lattices of alpha-Meet-Closed Subsets
Abstract
We study abstract properties of intervals in the complete lattice of all kappa-meet-closed subsets (kappa-subsemilattices) of a kappa-(meet-)semilattice S, where kappa 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 boolean OR {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 kappa-subsemilattices, we describe the covering relation, the coatoms, the boolean OR-irreducible and the boolean OR-prime elements in terms of the underlying kappa-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.
Year
DOI
Venue
2004
10.1007/s11083-004-3716-2
ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS
Keywords
DocType
Volume
(weakly) atomic,(strongly) coatomic,complete lattice,extremally detachable,interval,irreducible,meet-closed,prime,semilattice
Journal
21
Issue
ISSN
Citations 
2
0167-8094
0
PageRank 
References 
Authors
0.34
2
1
Name
Order
Citations
PageRank
Marcel Erné12910.77