Generalized homothetic biorders

In this paper, we study the binary relations RR on a nonempty N∗N∗-set AA which are h-independent and h-positive (cf. the introduction below). They are called homothetic positive orders  . Denote by BB the set of intervals of RR having the form [r,+∞[[r,+∞[ with 00Q>0-set endowed with a binary relation >> extending the usual one on R>0R>0 (identified with a subset of BB via the map r↦[r,+∞[r↦[r,+∞[). We first prove that there exists a unique   map ΦR:A×A→BΦR: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)Φ(mx,ny)=mn−1⋅Φ(x,y) and xRy⇔ΦR(x,y)>1. Then we give a characterization of the homothetic positive orders RR on AA such that there exist two morphisms of N∗N∗-sets u1,u2:A→B satisfying xRy⇔u1(x)>u2(y). They are called generalized homothetic biorders  . Moreover, if we impose some natural conditions on the sets u1(A)u1(A) and u2(A)u2(A), the representation (u1,u2)(u1,u2) is “uniquely” determined by RR. For a generalized homothetic biorder RR on AA, the binary relation R1R1 on AA defined by xR1y⇔ΦR(x,y)>ΦR(y,x) is a generalized homothetic weak order  ; i.e. there exists a morphism of N∗N∗-sets u:A→Bu:A→B such that (for all x,y∈A) we have xR1y⇔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)(u1,u2) of RR in terms of uu and a twisting factor.