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 a1 , a2 , a3 , ..., an like that a1 ≤ a2 , a2 ≤ a3 , ..., an-1 ≤ an and an ≤ a1 . Applying the transitivity n-1 times on a1 ≤ a2 , a2 ≤ a3 , ..., an-1 ≤ an , we get a1 ≤ an .As relation ≤ is anti-symmetric a1 ≤ an and an ≤ a1 together entails that a1 = an . This is contrary to the fact that all a1, a2, a3... an are distinct. So, our assumption that there is a cycle of length n ≥ 2 in the digraph of a partial order relation is wrong.