Strict partial order

Remarks

The relation "is a proper subset of" on the power set of a set is a strict partial order. It is irreflexive since no set is a proper subset of itself, and it is transitive since $X \subsetneq Y \subsetneq Z$ gives $X \subsetneq Z$. The relation $\ne$ on $\R$ is irreflexive but not transitive: $1 \ne 2$ and $2 \ne 1$, but $1 \ne 1$ fails. A strict partial order is asymmetric: $a < b$ and $b < a$ together would give $a < a$. From a partial order $\le$ one gets a strict partial order by $a < b$ if and only if $a \le b$ and $a \ne b$. Conversely, from a strict partial order one gets a partial order by $a \le b$ if and only if $a < b$ or $a = b$. These two constructions are inverse to each other.