Definition 4.21.1. Let $I$ be a set and let $\leq $ be a binary relation on $I$.
We say $\leq $ is a preorder if it is transitive (if $i \leq j$ and $j \leq k$ then $i \leq k$) and reflexive ($i \leq i$ for all $i \in I$).
A preordered set is a set endowed with a preorder.
A directed set is a preordered set $(I, \leq )$ such that $I$ is not empty and such that $\forall i, j \in I$, there exists $k \in I$ with $i \leq k, j \leq k$.
We say $\leq $ is a partial order if it is a preorder which is antisymmetric (if $i \leq j$ and $j \leq i$, then $i = j$).
A partially ordered set is a set endowed with a partial order.
A directed partially ordered set is a directed set whose ordering is a partial order.