  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., completely meet-irreducible) elements of Reg(e) and
the arrow relations between them. In particular, we prove that (1) Reg(e) is
the Dedekind-MacNeille completion of the poset Clop(e); (2) Every open subset
of e is a set-theoretic union of completely join-irreducible clopen subsets of
e; (3) Clop(e) is a lattice iiff every regular closed subset of e is clopen,
iff e contains no "square" configuration, iff Reg(e)=Clop(e); (4) If e is
finite, then Reg(e) is pseudocomplemented iff it is semidistributive, iff it is
a bounded homomorphic image of a free lattice, iff e is a disjoint sum of
antisymmetric transitive relations and two-element full relations. We
illustrate the strength of our results by proving that, for n greater than or
equal to 3, the congruence lattice of the lattice Bip(n) of all bipartitions of
an n-element set is obtained by adding a new top element to a Boolean lattice
with n2^{n-1} atoms. We also determine the factors of the minimal subdirect
decomposition of Bip(n).
