Let $A$ and $B$ be strings consisting of small latin alphabets.
We will say $A<B$ iff $AB$ is lexicography smaller than $BA$. ($AB$ is string concatenation of $A$ and then $B$. For example, $A="abcd"$, $B="xyz"$, $AB="abcdxyz"$)
Can we show that if $A<B$ and $B<C$, then $A<C$ (or show that it is wrong) ?