Basic question about partially ordered sets.

Question

Let $(X,\le)$ be a partially ordered set. Let $x,y,z \in X$. Suppose $x \le y$ and $x \le z$. Then are $y$ and $z$ comparable?

Answer 1

Not necessarily. Consider the set $X=\mathscr{P}(\{1,2,3\})$ partially ordered by $\subseteq$, then $\{1\}\subseteq\{1,2\}$ and $\{1\}\subseteq\{1,3\}$ but $\{1,2\}$ and $\{1,3\}$ are not comparable. What we do usually say in this case is that $\{1,2\}$ and $\{1,3\}$ are compatible.