Pre Order and Partial Order

Let $S$ be some set and $\leq$ be some relation between the elements of the set.

Pre-order Conditions¶

The relation sets a pre-order on the set $S$ if

1. $a \leq a$ for all $a~\in~S$
   • Reflexivity
2. $a \leq b$ and $b \leq c$ implies $a \leq c$ where $a,b,c~\in~S$.
   • Transitivity

Partial-order Conditions¶

The relation sets a partial order on the set $S$ if both 1 and 2 from the pre-order conditions are met and

3. $a \leq b$ and $b \leq a$ implies $a=b$
   • Antisymmetry

Total Ordering¶

If every element in $S$ can be compared by the relation, then the relation forms a total ordering on $S$.