Generalized homothetic biorders

In this paper, we study the binary relations R on a nonempty N-set A which are hindependent and h-positive (cf. the introduction below). They are called homothetic positive orders. Denote by B the set of intervals of R having the form [r,+∞[ with 0 < r ≤ +∞ or ]q,∞[ with q ∈ Q≥0. It is a Q>0-set endowed with a binary relation > extending the usual one on R>0 (identified with a subset of B via the map r 7→ [r,+∞[). We first prove that there exists a unique map ΦR : A × A → B such that (for all x, y ∈ A and all m, n ∈ N) we have Φ(mx, ny) = mn−1 · Φ(x, y) and x R y ⇔ ΦR(x, y) > 1. Then we give a characterization of the homothetic positive orders R on A such that there exist two morphisms of N-sets u1, u2 : A → B satisfying x R y ⇔ u1(x) > u2(y). They are called generalized homothetic biorders. Moreover, if we impose some natural conditions on the sets u1(A) and u2(A), the representation (u1, u2) is ‘‘uniquely’’ determined by R. For a generalized homothetic biorder R on A, the binary relation R1 on A defined by x R1 y ⇔ ΦR(x, y) > ΦR(y, x) is a generalized homothetic weak order; i.e. there exists a morphism of N-sets u : A → B such that (for all x, y ∈ A) we have x R1 y ⇔ u(x) > u(y). As we did in [B. Lemaire, M. Le Menestrel, Homothetic interval orders, Discrete Math. 306 (2006) 1669–1683] for homothetic interval orders, we also write ‘‘the’’ representation (u1, u2) of R in terms of u and a twisting factor. © 2008 Elsevier B.V. All rights reserved. This paper proposes a generalization of [13] in which we had studied homothetic interval orders on a nonempty N-set A. Let us recall that such an order R is a nonempty binary relation, h-independent in the sense that x R y ⇔ mx Rmy for all x, y ∈ A and all m ∈ N, and satisfying a series of properties that ensure the existence of two morphisms of N-sets u1, u2 : A→ R>0 such that x R y⇔ u1(x) > u2(y) with u1 ≤ u2. Moreover, the pair (u1, u2) is unique up to multiplication by a positive scalar. Besides h-independence, the most striking properties of homothetic interval orders are: – asymmetry: x R y⇒ y (−R) xwhere−Rmeans the negation of R; – h-positivity: for allm, n ∈ N such thatm > n, we have x R y⇒ mx R ny; – h-super-Archimedean1: if x R y, then there existsm ∈ N such thatmx R (m+ 1)y. Note that asymmetry implies – irreflexivity: x (−R) x. Of all these properties, this paper first retains only two: h-independence and h-positivity. I This paper has been announced in [M. Le Menestrel, B. Lemaire, Ratio-scale measurement with intransitivity or incompleteness: The homogeneous case, Theory Decis. 60 (2006) 207–217; B. Lemaire, M. Le Menestrel, Homothetic interval orders, Discrete Math. 306 (2006) 1669–1683] under the title of ‘‘Homothetic positive orders’’. ∗ Corresponding author. E-mail addresses: [email protected] (B. Lemaire), [email protected] (M. Le Menestrel). 1 In [13], we called this property h-Archimedean but the terminology of the present paper is more in line with the literature (see e.g. [6]). 0012-365X/$ – see front matter© 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2008.10.011 Please cite this article in press as: B. Lemaire, M. Le Menestrel, Generalized homothetic biorders, Discrete Mathematics (2008), doi:10.1016/j.disc.2008.10.011