quasiorders

Quasiorder, also called a preorder, is a binary relation ≤ on a set S that is reflexive (for every x in S, x ≤ x) and transitive (if x ≤ y and y ≤ z, then x ≤ z). It generalizes a partial order, with the key difference that antisymmetry is not required; that is, it may happen that x ≤ y and y ≤ x for distinct elements x ≠ y.

Given a quasiorder, define x ~ y if x ≤ y and y ≤ x. This is an equivalence relation that partitions S into equivalence classes. The set of classes S/~ inherits a partial order defined by [x] ≤ [y] if and only if x ≤ y. Thus every quasiorder induces a poset on the quotient, showing how preorders relate to partial orders.

Examples: The universal relation on any nonempty set is a quasiorder that is not antisymmetric when the set has more than one element. The natural numbers with the usual ≤ form a quasiorder and, in fact, a partial order. In topology, the specialization preorder on a space defines a quasiorder where antisymmetry holds exactly for T0 spaces; this links order-theoretic and topological structure.

Additional notes: Quasiorders can be viewed as thin categories, where a ≤ b corresponds to a unique morphism from a to b. They often arise as the reflexive transitive closure of a directed relation and are used in computer science to model refinement, information ordering, and epistemic or observational equivalence where distinct elements may be indistinguishable under the relation.