Different definitions of the concept of a fuzzy semiorder are compared. It is proved that their α-cuts are crisp binary relations that may fail to be Ferrers and semitransitive, in general. Consequently, we analyze the preservation of semiorders when coming back from the fuzzy to the crisp setting using α-cuts. In the final sections, a discussion is developed about the extension to the fuzzy setting of the concept of a threshold of utility discrimination, and its corresponding numerical representability of fuzzy semiorders by means of the representability of their α-cuts as crisp binary relations.