Details
| Original language | English |
|---|---|
| Pages (from-to) | 137-153 |
| Number of pages | 17 |
| Journal | ORDER |
| Volume | 21 |
| Issue number | 2 |
| Publication status | Published - 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
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Geometry and Topology
- Computer Science(all)
- Computational Theory and Mathematics
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In: ORDER, Vol. 21, No. 2, 18.08.2004, p. 137-153.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Intervals in lattices of κ-meet-closed subsets
AU - Erné, Marcel
PY - 2004/8/18
Y1 - 2004/8/18
N2 - 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.
AB - 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.
KW - (Strongly) coatomic
KW - (Weakly) atomic
KW - Complete lattice
KW - Extremally detachable
KW - Interval
KW - Irreducible
KW - Meet-closed
KW - Prime
KW - Semilattice
UR - http://www.scopus.com/inward/record.url?scp=24344477065&partnerID=8YFLogxK
U2 - 10.1007/s11083-004-3716-2
DO - 10.1007/s11083-004-3716-2
M3 - Article
AN - SCOPUS:24344477065
VL - 21
SP - 137
EP - 153
JO - ORDER
JF - ORDER
SN - 0167-8094
IS - 2
ER -