A partial ordering on a Set $X$ is a (binary) Relation on $X$, $\preceq$ that is:

• reflexive: for all $x \in X, x \preceq x$.
• antisymmetric: for all $x, y \in X$, if $x \preceq y$, and $y \preceq x$, then $x=y$.
• transitive: for all $x, y, z \in X$, if $x \preceq y$ and $y \preceq z$, then $x \preceq z$/

A set with a partial ordering is called a Partially ordered set (or poset).

A Pre-order is a weaker kind of relation

For many common examples, the Partial ordering $\preceq$ is often interpreted as $\leq$ (or less than or equal, for Real numbers).