Let $S$ be a semigroup with zero and $E(S)$ be the set of all idempotent of $S$. The definition of primitiveidempotent has written in the book " Fundaentals of Semigroup Theory" by "James Howie". The idempotents that are minimal within the set of nonzero idempotents are called primitive. Here partial oreder relation on $E(S)$ is $e \leq f$ if $ef = fe = e.$
My question is:
$0$ is minimal idempotent . Can we consider $0$ is primitive element of $S$.
I would be Thankful if someone help me.