Irreflexive relation ($\forall x\neg R(x,x)$)

A binary relation is irreflexive when no element is related to itself. An irreflexive relation that is also a transitive relation is a strict partial order. Transitivity then excludes every finite directed cycle, since a cycle would imply $R(x,x)$ at its starting element.

A binary relation is irreflexive exactly when it contains no diagonal pair $(x,x)$. Its negation requires at least one diagonal pair, whereas a reflexive relation requires every diagonal pair.