Intervals in Lattices of -Meet-Closed Subsets

We study abstract properties of intervals in the complete lattice of all meet-closed subsets ( -subsemilattices) of a -(meet-)semilattice S, where 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 closed set generated by A and x leaves a closed set). Such closure systems have many pleasant geometric and latticetheoretical 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 -subsemilattices, we describe the covering relation, the coatoms, the W -irreducible and the W -prime elements in terms of the underlying -semilattices. Although such intervals may fail to be lower continuous, they are strongly coatomic iff every element has an irredundant (and even a least) join-decomposition. We also characterize those intervals which are Boolean, distributive (equivalently: modular), or semimodular, respectively. Mathematics Subject Classification: primary: 06A12, secondary: 06B05, 06A23, 52A01.