Strict partial order

A strict partial order on a set $P$ is a relation $<$ that is irreflexive, meaning $a\not<a$ for every $a$ , and transitive, meaning $a<b$ and $b<c$ imply $a<c$ . These properties imply asymmetry: $a<b$ excludes $b<a$ .

Nonstrict form

Setting $a\le b$ when $a=b$ or $a<b$ gives a partial order. Conversely, deleting the diagonal from a partial order gives a strict partial order. Comparability of every distinct pair is not required. A strict partial order on an infinite set may have infinite descending chains.