Set/Order relation/Definition

A relation ${\displaystyle {}\preccurlyeq }$ on a set ${\displaystyle {}I}$ is called an order relation if the following three conditions are fulfilled.

1. ${\displaystyle {}i\preccurlyeq i}$ for all ${\displaystyle {}i\in I}$.
2. From ${\displaystyle {}i\preccurlyeq j}$ and ${\displaystyle {}j\preccurlyeq k}$ we can deduce ${\displaystyle {}i\preccurlyeq k}$.
3. From ${\displaystyle {}i\preccurlyeq j}$ and ${\displaystyle {}j\preccurlyeq i}$ we can deduce ${\displaystyle {}i=j}$.