Complete PreOrder
This is a preference structure for Modeling Preferences in MCDA
Also called the traditional representation of preferences
$\forall a,b \in A:$
 $a \ P \ b \iff g(a) > g(b)$ (The complete order: $R$ relation)
 $a \ I \ b \iff g(a) = g(b)$ (The complete preoder: $I$ relation)
$g$ is some global aggregation function:
This way we cannot model incomparability ($>$ can always compare things)
 $J$ is always empty  everything is comparable
 $P$ is transitive
 $I$ also becomes transitive
Example 1
Suppose there are three sport teams: $a$, $b$, and $c$.
 If $a$ beats $b$, $a$ receives 3 points and the loser receives 0 points
 If they draw, both receives 1 point.
 The three teams will play with each other and at the end they will have total points.
 If all total scores are different, there will be a complete order
 If there is a tie, the order will be a complete preorder
Example 2
Expected gains of different actions:
$a$ 
$b$ 
$c$ 
$d$ 
$e$ 
$f$ 
$g$

100 
100 
120 
130 
130 
130 
131

Sources