  Every partial applicative structure gives rise to an indexed binary relation,
that is a contravariant functor from the category of sets to the category of
sets endowed with binary relations and maps preserving them. In this paper we
characterize those partial applicative structures giving rise to indexed
relations satisfying certain elementary properties in terms of algebraic or
computational properties. We will then provide a characterization of those
partial applicative structures giving rise to indexed preorders and indexed
posets, and we will relate the latter ones to some particular classes of unary
partial endofunctions. We will analyze the relation between a series of
computational and algebraic properties in the posetal case. Finally, we will
study the problem of existence of suprema in the case of partial applicative
structures giving rise to indexed preorders, by providing some necessary
conditions for a partial applicative structure to be complete.
