Homothetic interval orders

We give a characterization of the non-empty binary relations  on a N∗-set A such that there exist two morphisms of N∗-sets u1, u2 : A → R+ verifying u1 ≤ u2 and x  y ⇔ u1(x) > u2(y). They are called homothetic interval orders. If  is a homothetic interval order, we also give a representation of  in terms of one morphism of N∗-sets u : A → R+ and a map σ : u(R+) × A → R+ such that x  y ⇔ σ(x, y)u(x) > u(y). The pairs (u1, u2) and (u, σ) are “uniquely” determined by Â, which allows us to recover one from each other. We prove that  is a semiorder (resp. a weak order) if and only if σ is a constant map (resp. σ = 1). If moreover A is endowed with a structure of commutative semigroup, we give a characterization of the homothetic interval orders  represented by a pair (u,σ) so that u is a morphism of semigroups. Résumé On donne une caractérisation des relations binaires non vides  sur un N∗-ensemble A telles qu’il existe deux morphismes de N∗-ensembles u1, u2 : A → R+ vérifiant u1 ≤ u2 et x  y ⇔ u1(x) > u2(y). On les appelle des ordres intervalles homothétiques. Si  est un ordre intervalle homothétique, on donne aussi une représentation de  à l’aide d’un morphisme de N∗-ensembles u : A → R+ et d’une application σ : u(R+)× A → R+ tels que x  y ⇔ σ(x, y)u(x) > u(y). Les paires (u1, u2) et (u,σ) sont déterminées “de manière unique” par Â, ce qui nous permet de retrouver l’une à partir de l’autre. On montre que  est un semiordre (resp. un ordre faible) si et seulement si σ est une application constante (resp. σ = 1). Si de plus A est muni d’une structure de semigroupe commutatif, on donne une caractérisation des ordres intervalles homothétiques  représentés par une paire (u,σ) telle que u soit un morphisme de semigroupes. AMS Class. 06A06, 06F05, 20M14 Key-words N∗-set, semigroup, weak order, semiorder, interval order, intransitive indifference, independence, homothetic structure, representation. Introduction Let us start with an example, which has been our main source of inspiration for this work. Consider a two-armed-balance, the two arms of which not necessarily being of the same length; such a balance is said to be biased. Let denote P1 and P2 its two pans. If the arms are not of the same length, we assume that P1 is located at the end of the shortest arm. Suppose also we are given a set A of objects to put on P1 and P2. We define as follows a binary relation  on A: x  y if the balance tilts towards x when we put x on P1 and y on P2. This relation is always asymmetric and transitive, but it is negatively transitive if and only if the two arms are of the same length. However we can observe it is always strongly transitive: x  y % z  t ⇒ x  t with y % z ⇔ z 6 y. In particular,  is an interval order (cf. [F]). Furthemore, suppose that A is endowed with a structure of N∗-set. Then the relation  verifies the following property of homothetic independence: x  y ⇔ (mx  my, ∀m ∈ N∗). We can continue to identify the properties satisfied by Â. That naturally brings us to introduce the notion of homothetic structure (cf. section 2). A homothetic structure is by definition a N∗-set A endowed with a binary relation  verifying five properties of compatibility, the most striking two being the homothetic independence 1 UMR 8628 du CNRS, Département de Mathématiques de l’Université de Paris-Sud, bâtiment 425, 91405 Orsay cedex France; e-mail: [email protected] 2 Universitat Pompeu Fabra, Departament d’Economia i Empresa, Ramon Trias Fargas 25-27, 08005-Barcelona (Espanya); e-mail: [email protected]