A preorder on a set S is a relation ≤ that is reflexive (a ≤ a for all a) and transitive (a ≤ b and b ≤ c imply a ≤ c).
A preorder is the weakest structure that supports a notion of “at least as much as.” It allows ties: elements a and b can satisfy both a ≤ b and b ≤ a without being equal. When ties are ruled out (by adding antisymmetry), the preorder becomes a partial order.
Every preorder on S can be viewed as a thin category: objects are elements of S, and there is a unique morphism from a to b whenever a ≤ b. Reflexivity provides identity morphisms; transitivity provides composition.
In modal logic, Kripke frames for S4 are preorders: the accessibility relation is reflexive and transitive. The elements (possible worlds) are ordered by “accessibility” — later worlds see at least as much as earlier ones — and the modal operators quantify over this order.
The equivalence relation induced by a preorder (a ~ b when a ≤ b and b ≤ a) partitions S into equivalence classes. The quotient of a preorder by this relation is a partial order — antisymmetry is gained by identifying tied elements.