Prefix relation on words in $\Sigma^*$ - why does a maximum element imply that the prefix relation is a linear order?

Question

I'm currently preparing for a test, and I'm having trouble understanding one of the preparation questions. The question is as follows:

Let $\Sigma$ be a finite alphabet. The prefix relation on words in $\Sigma^*$ is a binary relation $<_{pref}$ such that:

$x <_{pref} y$ if and only if $\exists z \in \Sigma^* : xz = y$

Prove that if on a language $L \subset \Sigma^*$, $<_{pref}$ has a maximum element, then $<_{pref}$ is a linear order.

My understanding is that the maximum element enforces totality (because that's the only thing separating a partial order from a linear one), but I can't understand exactly why. Any help would be greatly appreciated.

Answer 1

I think is simply by the definition of linear order or total order.

If you have a maximum element, is because you have a prefix in common, so you can prove the axioms of the definition on linear order..

http://en.wikipedia.org/wiki/Total_order