Condorcet domains and distributive lattices

Condorcet domains are sets of linear orders where Condorcet's effect can never occur. Works of Abello, Chameni-Nembua, Fishburn and Galambos and Reiner have allowed a strong understanding of a significant class of Condorcet domains which are distributive lattices -in fact covering distributive sublattices of the permutoèdre lattice- and which can be obtained from a maximal chain of this lattice. We describe this class and we study three particular types of such Condorcet domains.Acyclic set, alternating scheme, Condorcet effect, distributive lattice, maximal chain of permutations, permutoèdre lattice.

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., 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).Comment: 25 page

Condorcet domains and distributive lattices

URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSECahiers de la Maison des Sciences Economiques 2006.72 - ISSN 1624-0340Condorcet domains are sets of linear orders where Condorcet's effect can never occur. Works of Abello, Chameni-Nembua, Fishburn and Galambos and Reiner have allowed a strong understanding of a significant class of Condorcet domains which are distributive lattices -in fact covering distributive sublattices of the permutoèdre lattice- and which can be obtained from a maximal chain of this lattice. We describe this class and we study three particular types of such Condorcet domains.Un domaine condorcéen est un ensemble d'ordres totaux où la règle majoritaire s'applique sans «effet Condorcet» : la relation majoritaire de tout profil de préférences choisies dans cet ensemble n'admet aucun circuit. Des travaux d'Abello, Chameni-Nembua, Fishburn et Galambos et Reiner ont permis une compréhension profonde d'une classe de domaines condorcéens qui sont des treillis distributifs. Ce sont en fait, des sous-treillis distributifs couvrants du treillis permutoèdre et on les obtient à partir d'une chaîne maximale de ce treillis. Je décris cette classe et j'en étudie trois types particuliers importants

Acyclic domains of linear orders: a survey

Among the many significant contributions that Fishburn made to social choice theory some have focused on what he has called "acyclic sets", i.e. the sets of linear orders where majority rule applies without the "Condorcet effect" (majority relation never has cycles). The search for large domains of this type is a fascinating topic. I review the works in this field and in particular consider a recent one that allows to show the connections between some of them that have been unrelated up to now.acyclic set;alternating scheme;distributive lattice;effet Condorcet;linear order,maximal chain,permutoèdre lattice, single-peaked domain,weak Bruhat order,value restriction.

A characterization for all interval doubling schemes of the lattice of permutations

The lattice \textbfS_n of all permutations on a n-element set has been shown to be \emphbounded [CAS], which is a strong constructive property characterized by the fact that \textbfS_n admits what we call an \emph interval doubling scheme. In this paper we characterize all interval doubling schemes of the lattice \textbfS_n, a result that gives a nice precision on the bounded nature of the lattice of permutations. This theorem is a direct corollary of two strong properties that are also given with their proofs

Cayley lattices of finite Coxeter groups are bounded

AbstractAn interval doubling is a constructive operation which applies on a poset P and an interval I of P and constructs a new “bigger” poset P′=P[I] by replacing in P the interval I with its direct product with the two-element lattice. The main contribution of this paper is to prove that finite Coxeter lattices are bounded, i.e., that they can be constructed starting with the two-element lattice by a finite series of interval doublings.The boundedness of finite Coxeter lattices strengthens their algebraic property of semidistributivity. It also brings to light a relation between the interval doubling construction and the reflections of Coxeter groups.Our approach to the question is somewhat indirect. We first define a new class HH of lattices and prove that every lattice of HH is bounded. We then show that Coxeter lattices are in HH and the theorem follows. Another result says that, given a Coxeter lattice LW and a parabolic subgroup WH of the finite Coxeter group W, we can construct LW starting from WH by a series of interval doublings. For instance the lattice, associated with An, of all the permutations on n+1 elements is obtained from An−1 by a series of interval doublings. The same holds for the lattices associated with the other infinite families of Coxeter groups Bn, Dn and I2(n)

"Mathématique Sociale" and Mathematics. A case study: Condorcet's effect and medians

The "effet Condorcet" refers to the fact that the application of the pair-wise majority rule to individual preference orderings can generate a collective preference containing cycles. Condorcet's solution to deal with this disturbing fact has been recognized as the search for a median in a certain metric space. We describe the many areas of "applied" or "pure" mathematics where the notion of (metric) median has appeared. If it were actually necessary to give examples proving that "social mathematics" is mathematics, the median case would provide a convincing example.Condorcet's effect ; Fermat's point ; majority rule ; "Mathématique sociale" ; median algebra ; metric space ; permutohedron