Composition of linear orders on the same set

Question

I need to show an example of finite linear orders $K_1$ and $K_2$ are defined on the same set, such that their composition $K_1\circ K_2$ is not a transitive relation.

Whatever orders I choose, their composition is a transitive relation.

Any ideas? Thank you for your time.

Answer 1

Consider the 5-element set consisting of the five elements a, b, c, d, e. Let $K_1$ order these elements in the order e, c, d, a, b. Let $K_2$ order them in the order d, e, b, c, a. Under the composite relation $K_1\circ K_2$, transitivity fails for the triple a, c, e.

Answer 2

There is a general result: if $K$, $L$ are transitive relations and $KL=LK$ then $KL$ is transitive.