A preorder on a set S is a binary relation $\leq$ that is reflexive ($a \leq a$ for all a) and transitive ($a \leq b$ and $b \leq c$ imply $a \leq c$).

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 $a \leq b$ and $b \leq a$ without being equal. When ties are ruled out by adding antisymmetry, the preorder becomes a partial order.

Every preorder induces an equivalence relation: define $a \sim b$ when $a \leq b$ and $b \leq a$. This partitions S into equivalence classes of tied elements. The quotient of a preorder by this relation is a partial order — antisymmetry is gained by identifying tied elements. Every preorder is a partial order “up to ties.”

Every preorder on S can be viewed as a thin category: objects are elements of S, and there is a morphism from a to b whenever $a \leq b$. Reflexivity provides identity morphisms. Transitivity provides composition. A partial order is a thin category where there is at most one morphism between any two objects. A preorder is a thin category where there can be morphisms in both directions between distinct objects — the ties.

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 (necessity, possibility) quantify over this order. Reflexivity gives the T axiom (what is necessary is true). Transitivity gives the 4 axiom (what is necessary is necessarily necessary).