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 -