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.


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


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