  For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open
subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f),
extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite
convex geometry, then Reg(P,f) 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,f)
carries a particularly interesting structure for special types of convex
geometries, that we call closure spaces of semilattice type. For finite such
closure spaces, (1) Reg(P,f) satisfies an infinite collection of stronger and
stronger quasi-identities, weaker than both meet- and join-semidistributivity.
Nevertheless it may fail semidistributivity. (2) If Reg(P,f) is
semidistributive, then it is a bounded homomorphic image of a free lattice. (3)
Clop(P,f) is a lattice iff 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 regular closed sets is, in the
semilattice context, always the Dedekind Mac-Neille 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.
