# Is there a name for a semigroup whose idempotents form a subsemigroup?

For a semigroup $S,$ I will denote by $E(S)$ the set of all idempotents of $S$. For $X\subseteq S,$ let $X^2$ mean $\{xy\,|\,x,y\in X\}.$

Is there a name for the class of semigroups $S$ such that $$\left(E(S)\right)^2\subseteq E(S)?$$

To have an example, in every inverse semigroup, the idempotents form a subsemigroup. More generally, as rschwieb points out in a comment, any semigroup such that the idempotents commute with each other satisfies this condition.

I need a name to be able to search for information about such semigroups. So any contribution besides the name will be welcomed.

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We can start with abelian semigroups meaning "commutative". In ring theory sometimes "abelian" is used to mean "central idempotents", and that would work here too. –  rschwieb Jun 28 '12 at 20:48
@rschwieb Do you mean that idempotents form a subsemigroup iff they commute with everything? –  user23211 Jun 28 '12 at 20:55
No, sorry, I did not even attempt to classify, only to give more examples. I mean that if idempotents commute with each other, then they are obviously closed under multiplication: $efef=eeff=ef$ if $e,f$ are idempotents. –  rschwieb Jun 28 '12 at 20:56
@rschwieb Oh, OK. Yes, I know about it -- I should have put it in the question. This is actually the case in inverse semigroups: an inverse semigroup is a regular semigroup whose idempotents commute. –  user23211 Jun 28 '12 at 21:00
I've just found out that the answer to my question in the comment is no: in the bicyclic semigroup, the idempotents do not commute with everything. –  user23211 Jun 28 '12 at 21:01
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According to wikipedia, we have

• A regular semigroup whose idempotents forms a subsemigroup is called an orthodox semigroup.
• A completely regular semigroup whose idempotents forms a subsemigroup is called an orthogroup.

Every semigroup can be embedded into a regular semigroup. Perhaps subsemigroup of an orthodox semigroup comes closest to the condition you are looking for. I don't know whether there exists a semigroup which satisfies your condition without being a subsemigroup of an orthodox semigroup.

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Thank you, Thomas. I haven't seen these terms before, so this is very helpful. I will accept this answer if no full answer comes up. –  user23211 Jun 29 '12 at 22:56