Preorder

A preorder is a reflexive and transitive relation, without any requirement of symmetry or antisymmetry. It generalizes both equivalence relations and partial orders.

Binary relation: A relation between elements of two sets, represented as a set of ordered pairs. A preorder is a binary relation on a set whose pairs obey two additional laws.
Partial order: A reflexive, antisymmetric, and transitive relation. Adding antisymmetry to a preorder produces a partial order.
Preorder quotient: The partial order formed by identifying elements related in both directions within a preorder. This construction removes the preorder's failure of antisymmetry.
Domain theory: The mathematical study of ordered structures used to model computation and recursive definitions. Preorders can represent approximation when distinct descriptions count as mutually comparable.
Reflexive relation: A relation in which every element is related to itself. Reflexivity is one of the two defining conditions of a preorder.
Equivalence relation: A reflexive, symmetric, and transitive relation. Adding symmetry to a preorder produces an equivalence relation.
Equivalence class: The set of elements equivalent to a given element under an equivalence relation. Mutual comparison in a preorder defines the equivalence classes used in its quotient.
Information ordering: An order expressing when one information state contains no more information than another. Preorders capture information growth even when different states convey the same information.
Transitive relation: A relation in which relating one element to a second and the second to a third implies relating the first to the third. Transitivity is the other defining condition of a preorder.
Quasiorder: A reflexive and transitive relation; usage sometimes distinguishes it from a preorder by convention. Quasiorder is a closely related name, though authors differ on whether it is synonymous.

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