The extended permutohedron on a transitive binary relation

For a given transitive binary relation  e on a set  E, the transitive closures of open (i.e., co-transitive in  e) sets, called the regular closed subsets, form an ortholattice Reg(e), the extended permutohedron on e. This construction, which contains the poset Clop(e) of all clopen sets, is a common generalization of known notions such as the generalized permutohedron on a partially ordered set on the one hand, and the bipartition lattice on a set on the other hand. We obtain a precise description of the completely join-irreducible (resp., meet-irreducible) elements of Reg(e) and the arrow relations between them. In particular, we prove that