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I am currently studying some basic facts of regular semigroups and Green's relations and I got stuck on the following exercise problem.

Let $S$ be a regular semigroup with a primitive idempotent $e$. Show, that $J(e)$ is the least ideal of $S$.

I will appreciate any solutions or ideas. Thanks! :)

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This is not true. Take the two-element monoid $\{1, 0\}$. Then $1$ is a primitive idempotent but the $\mathcal{J}$-class of $1$ is not even an ideal.

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  • $\begingroup$ Thank you. Would the statement hold if semigroup $S$ had no zero? $\endgroup$
    – Jiri Sedlacek
    Aug 29, 2013 at 10:23
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    $\begingroup$ If $S$ has no zero, then primitive means minimal for the natural order on idempotents. Thus the ideal generated by $e$ is now the $\mathcal{J}$-class of $e$ and this ideal is the unique minimal ideal of $S$. $\endgroup$
    – J.-E. Pin
    Aug 29, 2013 at 10:44

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