Can we have a Non-Reflexive Preference Relation? I've been thinking about preferences alot recently and have been specifically thinking about the reflexivity requirement.

That is:
$$x \succsim x$$

Though this is apparent and obvious, I have been wondering why this is a required condition for rationality and if its possible to have a preference relation that is complete but non-reflexive.

Is it possible?
Rationality requires the following:

Completeness

For all $x, y \in X$, either $x \succsim y$ or $y \succsim x$ or both.

Transitivity

For all $x, y, z \in X$, if $x \succsim y$ and $y \succsim z$, then $x \succsim z$.

Also note that if $x=y$ then completeness implies that $x\succsim x$. So reflexive preferences follow from completeness. So I would say Completeness and Transitivity are required for rationality. Reflexive preferences just follow from complete preferences.