Given a set $P$ endowed with a partial order $\preceq$, $x,y\in P$ are said to be comparable if and only if either $x\preceq y$ or $y\preceq x$.
Is comparability transitive? It would seem to be, but I see no mention of it on Wikipedia, except that incomparability is transitive for strict weak ordering.
I mean let's say $x,y$ and $y,z$ are pairwise comparable. Then clearly $x,z$ are comparable if ($x\preceq y$ and $y\preceq z$) or ($z\preceq y$ and $y\preceq x$), but what if $x\preceq y$ and $z\preceq y$? Is there a counterexample or proof?
Indeed, comparability is not necessarily transitive.
In the poset of subsets of $\{0,1\}$ ordered by inclusion, $\{0\}$ and $\{0,1\}$ are comparable, as are $\{0,1\}$ and $\{1\}$, but $\{0\}$ and $\{1\}$ are incomparable.