Prove that any directed cycle in the graph of a partial order must only involve one node
I'm assuming your partial orders are of the type "$a \leq b$" rather than of the type "$a &lt; b$", though the proof is very similar to both cases. Each edge $a \to b$ in the graph corresponds to an inequality $a \leq b$. Now suppose you had a triangle $a \to b \to c \to a$. Such a triangle would imply
$$a \leq b \leq c \leq a, $$
which has some implications since $\leq$ is a partial order. The case of longer (or shorter) cycles is similar.