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 x , y ∈ A and all m , n ∈ N ∗ ) we have Φ ( m x , n y ) = m n − 1 ⋅ Φ ( x , y ) and x R y ⇔ Φ 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 x R y ⇔ u 1 ( 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 x R 1 y ⇔ Φ R ( x , y ) > Φ 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 ∈ A ) we have x R 1 y ⇔ 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.