$\mathcal{J}-$ trivial elements

Question

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!

Answer 1

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?