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.