We give a characterization of the non-empty binary relations ≻ on a N * -set A such that there exist two morphisms of N * -sets u 1 , u 2 : A → R + verifying u 1 ⩽ u 2 and x ≻ y ⇔ u 1 ( x ) > u 2 ( 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 ( u 1 , u 2 ) 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.