Let $(P,\le)$ be a preorder, i.e. $P$ is a set and $\le$ is a relation on it that is reflexive and transitive. In this context for myself I can find two interpretations for the symbol $<$
1) $a<b$ iff $a\le b\wedge a\neq b$.
2) $a<b$ iff $a\le b\wedge\neg b\le a$
At first sight 1) seems normal. But looking at the partially ordered set that serves as skeleton of the preorder I would prefer 2).
Are there any conventions in order-theory about this notation?
It's the second.
(1) turns out to be a pretty bad relation; it's usually neither antisymmetric nor transitive!
You can fix (1) by changing it to $a \not\equiv b$, where $a \equiv b$ is defined to be $a \leq b \wedge b \leq a$.