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.
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Mathematics Subject Classifications (2000)
Primary: 06A12; Secondary: 06B05, 06A23, 52A01.
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Erné, M. Intervals in Lattices of κ-Meet-Closed Subsets. Order 21, 137–153 (2004). https://doi.org/10.1007/s11083-004-3716-2
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DOI: https://doi.org/10.1007/s11083-004-3716-2