Let $A = (Q,\Sigma, \delta)$ a finite complete deterministic automaton. Let's call $image$ the set $Qs$ for some word $ s \in \Sigma $ .
I'm wondering if the following definitions are equivalent:
DEF 1: a minimal image is an image Qs that doesn't properly include any other image.
DEF 2: A F-clique is a set Qs for some word $ s \in \Sigma^*$ such that, for every state $p, q\in Qs$ and for every $w \in \Sigma^*$, $pw \neq qw $
It's easy to show that an F-clique is a minimal image, but I'm wondering if the opposite is also true: is a minimal image always an F-clique?
The answer is yes. Let $Qs$ be a minimal image and let $w \in \Sigma^*$. If $pw = qw$ for some $p,q \in Qs$, then $|Qsw| < |Qs|$. It follows that $|Qsws| \leqslant |Qsw| < |Qs|$ and thus $Qsws$ is a proper subset of $Qs$, a contradiction. Thus $Qs$ is an $F$-clique.