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,+∞ 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?r,+∞). We first prove that there exists a unique map Φ_R:A×A→B such that (for all and all ) we have Φ(mx,ny)=mn⁻¹⋅Φ(x,y) and . Then we give a characterization of the homothetic positive orders R on A such that there exist two morphisms of N^∗-sets satisfying . 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 is a generalized homothetic weak order; i.e. there exists a morphism of N^∗-sets u:A→B such that (for all ) we have . 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.