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Closure lattices. (English) Zbl 0852.06002

Summary: Closure spaces have been previously investigated by Paul Edelman and Robert Jamison as ‘convex geometries’. Consequently, a number of the results given here duplicate theirs. However, we employ a slightly different, but equivalent, defining axiom which gives a new flavor to our presentation.
The major contribution is the definition of a partial order on all subsets, not just closed (or convex) subsets. It is shown that the subsets of a closure space, so ordered, form a lattice with regular, although non-modular, properties. Investigation of this lattice becomes our primary focus.

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
06A07 Combinatorics of partially ordered sets
05B35 Combinatorial aspects of matroids and geometric lattices
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