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.

## Comments (4)

Comment #2102 by Keenan Kidwell on

Comment #2129 by Johan on

Comment #2138 by Keenan Kidwell on

Comment #2146 by Johan on

There are also: