The author of the book I am studying defines $<$ for a poset as
If $x, y \in X$, where $X$ is a poset, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$.
From this, I can conceive of two definitions for $<$ for a preorder:
1) If $x, y \in X$, where $X$ is a preorder, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$.
or
2) If $x, y \in X$, where $X$ is a preorder, then we shall write $x < y$ to mean that $x \le y$ and $y \not\le x$.
Which of these is more appropriate?