1. The extended permutohedron on a transitive binary relation
- Author
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Luigi Santocanale, Friedrich Wehrung, Laboratoire d'informatique Fondamentale de Marseille - UMR 6166 ( LIF ), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Nicolas Oresme ( LMNO ), Université de Caen Normandie ( UNICAEN ), Normandie Université ( NU ) -Normandie Université ( NU ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire d'informatique Fondamentale de Marseille - UMR 6166 (LIF), Université de la Méditerranée - Aix-Marseille 2-Université de Provence - Aix-Marseille 1-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
bipartition ,orthocomplementation ,[ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM] ,Lattice (group) ,bipartite ,0102 computer and information sciences ,Disjoint sets ,join-dependency ,[ MATH.MATH-CO ] Mathematics [math]/Combinatorics [math.CO] ,01 natural sciences ,Combinatorics ,clopen ,[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] ,Clopen set ,open ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,transitive ,closed ,clepsydra ,0101 mathematics ,permutohedron ,Cambrian lattice ,Mathematics ,lattice ,Discrete mathematics ,relation ,Transitive relation ,Permutohedron ,Binary relation ,associahedron ,010102 general mathematics ,join-irreducible ,semidistributive ,regular closed ,Mathematics::Logic ,Poset ,square-free ,06A15, 05A18, 06A07, 06B10, 06B25, 20F55 ,010201 computation theory & mathematics ,subdirect product ,bounded ,Free lattice ,Combinatorics (math.CO) ,Partially ordered set - Abstract
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)., 25 pages
- Published
- 2014