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What does it mean by saying that

… idempotents are $\mathcal{J}-$ trivial.

Indeed, by searching old docs and this one, we see that:

A semigroup $S$ is $\mathcal{J}-$ trivial if two elements of $S$ which are $\mathcal{J}-$ equivalent are equal.

Is the following correct:

$a,b∈ S$ which $[a]_{\mathcal{J}}=[b]_{\mathcal{J}}$ are $\mathcal{J}-$ trivial in $S$ iff $a=b$.

Thanks for your hints!

Mikasa
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1 Answers1

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A semigroup is indeed $\mathcal{J}$-trivial is the Green's relation $\mathcal{J}$ is the equality. However, I do not understand the expressions "$a$ and $b$ are $\mathcal{J}$-trivial" and "idempotents are $\mathcal{J}$-trivial". Do you mean "the semigroup generated by the idempotents is $\mathcal{J}$-trivial" or something like that?

J.-E. Pin
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  • May I ask you: Is there any of your old works in which you did some facts related to ideals of a semigroup? – Mikasa Sep 14 '17 at 19:24
  • Semigroup ideals are a very basic notion in semigroup theory as well as the Green's preorder $\leqslant_\mathcal{J}$ and the Green's equivalence $\mathcal{J}$. They appear in every textbook in semigroup theory and they are used all the time (including in my own research). But you should start with the notions of minimal and $0$-minimal ideals, on which there are classical (but nontrivial) results. This material is covered in most books on semigroups. – J.-E. Pin Sep 15 '17 at 12:39