Motivation for this question
I'm studying preference relation (at MWG - Microeconomic Theory), and there's a relation $\succ$ (strict preference relation) which is defined as above:
$x \succ y \iff x\succsim y \land y \not \succsim x$
Where $\succsim$ (prefence relation) is a complete and trasitive relation.
It's easy to prove that $\succ$ relation is transitive (follows from the transitivity of $\succsim$) and antisymmetric (it is vacuous truth, because there's never a pair $(x,y)$ such that $x \succ y \land y \succ x$)
For me looks like $\succ$ relation is very similar to a partial order relation, but it's not reflexive. So this question came to my mind.
Why does the partial order relation needs to be reflexive?
What's the idea behind that? What results can be derived from the reflexive property that makes it relevant?
It's a bit strange for me because these two relations (a partial order and $\succ$) can be used to order some subsets from the original set.