Preorder

A (reflexive) preorder relation $\rho$ on set $A$ is a binary relation $r \subseteq A^2$ that is reflexive and transitive, that is, these two properties hold:

$x \mathop{\rho} x$

$x \mathop{\rho} y\ \land\ y \mathop{\rho} z \ \rightarrow\ x \mathop{\rho} z$

Constructing a partial order from a preorder

Intuitively, preorder differs from partial order in that there are distinct elements that have same ordering properties with respect to other elements. For such elements we therefore have $x \rho y$ and $y \rho x$ . By identifying these elements we obtain a partial order.

More precisely, let $\rho$ be a preorder. Define relation $\sim$ by

$\begin{equation*} x \sim y \ \iff\ x \mathop{\rho} y\ \land\ y \mathop{\rho} x \end{equation*}$

It is easy to verify that $\rho$ is an equivalence relation. Moreover, if we define relation $\leq$ on equivalence classes by

$\begin{equation*} P \leq Q\ \iff\ (\forall x \in P. \forall y \in Q. x \mathop{\rho} y) \end{equation*}$

for $P, Q \in A/_{\sim}$ , then we can prove that $\leq$ is a partial order.