Prove that the Digraph of a partial order has no cycle of length greater than 1.
Assume that there exists a cycle of length n ≥ 2 in the digraph of a partial order ≤ on a set A. This entails that there are n distinct elements a_{1} , a_{2} , a_{3} , ..., an like that a_{1} ≤ a_{2} , a2 ≤ a_{3} , ..., an-1 ≤ a_{n} and an ≤ a_{1} . Applying the transitivity n-1 times on a_{1} ≤ a_{2} , a_{2} ≤ a_{3} , ..., a_{n-1} ≤ a_{n} , we get a1 ≤ an .As relation ≤ is anti-symmetric a1 ≤ a_{n }and a_{n} ≤ a_{1} together entails that a_{1} = a_{n} . This is contrary to the fact that all a_{1,} a_{2}, a_{3}... a_{n} are distinct. So, our assumption that there is a cycle of length n ≥ 2 in the digraph of a partial order relation is wrong.