Following this book The Algebraic Theory of Semigroups, Volume I , we see that:
$a\mathcal{L}b$ means $a$ and $b$ generate the same principle left ideal of the semigroup $S$.
$a\mathcal{R}b$ means $a$ and $b$ generate the same principle right ideal of the semigroup $S$.
$\mathcal{D}$ is denoted for the join of the equivalence relations $\mathcal{L}$ and $\mathcal{R}$.
$\mathcal{H}$ is denoted for the intersection of the equivalence relations $\mathcal{L}$ and $\mathcal{R}$ and,
$a\mathcal{J}b$ means that $$S^1aS^1=S^1bS^1$$
I am asking if there is a name for the last equivalence relation as we have for other four relations. In fact, I am looking for a bold version for the last one. Thanks so much for your hints and the time.
I think what you are looking for is:
$a\mathcal{J}b$ means that $a$ and $b$ generate the same principal 2-sided ideal of the semigroup $S$.
For finite, or more generally periodic semigroups, $\mathcal{D}=\mathcal{J}$.