In order theory ideals are generally defined for partially ordered sets. Preorders (aka quasiorders) need not be anti-symmetric. They're reflexive and transitive.
Can I extend the notion of an ideal to preordered sets? That is, can I define a subset S of a preordered set P such that S satisfies the axioms of an ideal on a poset (i.e. a non-empty, directed, and lower set) and call S an ideal?