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 – e ) is the Dedekind–MacNeille completion of the poset Clop ( e ) ; open subset of  e is a set-theoretical union of completely join-irreducible clopen subsets of  e ; ( e ) is a lattice iff every regular closed subset of  e is clopen, iff  e contains no “square” configuration, iff Reg ( e ) = Clop ( 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. lustrate our results by proving that, for n ≥ 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 n ⋅ 2 n − 1 atoms. We also determine the factors of the minimal subdirect decomposition of Bip ( n ) , and we prove that if n ≥ 3 , then none of them embeds into  Bip ( n ) as a sublattice.