Generalized homothetic biorders

In this paper, We Study the binary relations R on a nonempty N*-set A which are h-independent 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, + infinity] with 0 < r <= + infinity or [q, infinity] with q is an element of 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 bar right arrow [r. + infinity]). We first prove that there exists a unique map Phi(R) : A x A -> B such that (for all x, y is an element of A and all m, n E N*) we have Phi(mx. ny) = mn(-1), Phi(x,y) and xRy double left right arrow Phi(R)(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 u(1), u(2) : A -> B satisfying xRy double left right arrow u1(x) > u(2)(y). They are called generalized homothetic biorders. Moreover, if we impose some natural conditions on the sets u(1)(A) and u(2)(A), the representation (u(1), u(2)) is "uniquely" determined by R. For a generalized homothetic biorder R on A, the binary relation R-1 on A defined by xR(1) y double left right arrow Phi(R)(x,y) > Phi(R)(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 is an element of A) we have xR(1) y double left right arrow 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 (u(1), u(2)) of R in terms of u and a twisting factor. (C) 2008 Elsevier B.V. All rights reserved.