Scott–Suppes representability of semiorders: Internal conditions

We analyze the structure of a semiorder, paying attention to its representability through a real-valued function and a positive constant threshold (the so-called Scott–Suppes representation). We furnish a new set of sufficient conditions for the Scott–Suppes representability of semiorders. Unlike previous characterizations already introduced in the literature, these new conditions can be expressed directly in terms of the given semiordered structure.