A partial order on a set is a relation that is reflexive, antisymmetric and transitive.

If $a\preceq b$, we say that $a$ precedes or is below $b$ in the order.

Unlike an ordinary numerical order, two elements need not be comparable. For example, set inclusion forms a partial order because neither of two different sets must contain the other.

Chains and incomparable elements

A chain is a subset whose elements are mutually comparable. Elements $a$ and $b$ are incomparable when neither $a\preceq b$ nor $b\preceq a$.

Hasse diagrams

For a finite partial order, a Hasse diagram draws only the immediate ordering relations. Reflexive edges and relations implied by transitivity are omitted.

Partial orders model dependency, precedence, refinement and containment without forcing unrelated objects into an artificial total order.