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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.

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    You do not know that $0$ exists. It is not requires for a structure to be called a semigroup. By the way, what even is $0$ if you have a multiplicative semigroup and not a ring (or rng, or rig, or any other similar structure)? – 2017-01-07
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    $S$ is a semigroup with zero with respect to the multiplication. Here $0$ means $0x= x0 = 0$ for all $x$ in $S$. – 2017-01-07
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    I already answered this question in a comment to this [question](http://math.stackexchange.com/questions/2080141/rees-theorem-on-semigroup-theory). Once again, since a primitive idempotent is by definition minimal within the set of nonzero idempotents, it is in particular a nonzero idempotent. – 2017-01-07
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    @J.-E.Pin: Assuming there is a minimal idempotent, is it possible to have several minimal idempotents in a finite semigroup? – 2017-09-08
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    Comments are not the right place to ask separate questions. – 2017-09-08

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