Lattices of regular closed subsets of closure spaces
Abstract
For a closure space (P, φ) with φ(ø) = ø, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P, φ), extending the poset Clop(P, φ) of all clopen subsets. If (P, φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind–MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P, φ) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For finite such closure spaces,
• Reg(P, φ) satisfies an infinite collection of stronger and stronger quasi-identities, weaker than both meet- and join-semidistributivity. Nevertheless it may fail semidistributivity.
• If Reg(P, φ) is semidistributive, then it is a bounded homomorphic image of a free lattice.
• Clop(P, φ) is a lattice if and only if every regular closed set is clopen.
The extended permutohedron R(G) on a graph G and the extended permutohedron Reg S on a join-semilattice S, are both defined as lattices of regular closed sets of suitable closure spaces. While the lattice of all regular closed sets is, in the semilattice context, always the Dedekind–MacNeille completion of the poset of clopen sets, this does not always hold in the graph context, although it always does so for finite block graphs and for cycles. Furthermore, both R(G) and Reg S are bounded homomorphic images of free lattices.
