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-1(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 ≻, 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.