Preorder (quasi-order)

Parent: basic-orders

Preorder

A relation ≤ that is reflexive (a ≤ a) and transitive (a≤b≤c ⟹ a≤c). Nothing else.

Because antisymmetry is not required, a preorder can rate two distinct elements ≤ both ways (a≤b and b≤a, a≠b) — they're "tied."

Preorder = equivalence + partial order. The "tied" relation a ~ b ⟺ a≤b ∧ b≤a is an equivalence-relation; quotient by it and the preorder becomes a genuine partial-order on the classes. So every preorder splits canonically into these two halves.

Examples: reachability in a directed graph; divisibility on ℤ (2 | −2 and −2 | 2, yet 2 ≠ −2 — not antisymmetric); big-O domination of functions.