What does $\mathcal{J}$ stand for in Green relations? 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.
 A: 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, $\mathcal{D}=\mathcal{J}$. 
A: It's perhaps worth noting that $\mathcal{J}$ can also be described in terms of joins/compositions $\mathcal{X}\circ\mathcal{Y}$, where $a(\mathcal{X}\circ\mathcal{Y})b\Leftrightarrow\exists c\in S(a\mathcal{X}c\mathcal{Y}b)$.  Indeed, if we assume $S=S^1$ and consider the relations
\begin{align*}
a\leq_\mathcal{L}b&\quad\Leftrightarrow\quad Sa\subseteq Sb\qquad\text{and}\\
a\leq_\mathcal{R}b&\quad\Leftrightarrow\quad aS\subseteq bS
\end{align*}
then the only difference between $\mathcal{D}$ and $\mathcal{J}$ is the order in which we take compositions and symmetrizations $\mathcal{X}^\mathrm{sym}=\mathcal{X}\cap\mathcal{X}^{-1}$, where $a\mathcal{X}^{-1}b\Leftrightarrow b\mathcal{X}a$.  Specifically,
\begin{align*}
\mathcal{L}&=\leq_\mathcal{L}^\mathrm{sym},\\
\mathcal{R}&=\leq_\mathcal{R}^\mathrm{sym},\\
\mathcal{D}&=\leq_\mathcal{L}^\mathrm{sym}\circ\leq_\mathcal{R}^\mathrm{sym}\quad\text{and}\\
\mathcal{J}&=(\leq_\mathcal{L}\circ\leq_\mathcal{R})^\mathrm{sym}.
\end{align*}
