Semiorders with separability properties

We analyze different separability conditions that characterize the numerical representability of semiorders through a real-valued function and a strictly positive threshold. Any necessary and sufficient condition for the numerical representability of an interval order by means of two real-valued functions is proved to also characterize the Scott–Suppes representability of semiorders provided that a key additional condition of regularity with respect to sequences holds.

► The Scott–Suppes representability of semiorders is revisited.
► A wide variety of conditions are proved to characterize the representability.
► A key conjecture has been solved.
► Interval order separability equals Scott–Suppes semiorder separability.