I recently came across the idea of extending the well-known Cayley graph construction for semigroups and learned that the outcome does not have all the expected properties even for the nice classes of semigroups, such as inverse semigroups. In particular, according to many sources, for inverse semigroups, the strongly connected components of the (left) Cayley graph are more important than the Cayley graph itself. These components are known as Schutzenberger graphs.
Here are the Schutzenberger graphs of the bicyclic monoid with the presentation $\mathcal{P}=\langle a, a^{\star} \mid a^{\star}a=1\rangle$ 
Upon seeing this, I have two questions:
- Are Schutzenberger graphs of different semigroup elements always isomorphic?
- Given all the Schutzenberger graphs, can we reconstruct the Cayley graph?
The strongly connected components of the left Cayley graph of a monoid are its $\mathcal L$-classes. They are not always isomorphic, even for an inverse monoid. For instance, in the inverse monoid $\{1, a, a^2\}$ with $a^3 = a$, the two $\mathcal L$-classes are $\{1\}$ and $\{a, a^2\}$.
It is not possible to recover the left Cayley graph from the $\mathcal L$-classes, even for an inverse semigroup. For instance, in the two idempotent and commutative semigroups with zero (semilattices) $M_1 = \{1, a, 0\}$ and $M_2 = \{a, b, 0\}$, where $ab = 0$, each singleton is an $\mathcal L$-class, but the left Cayley graph of $M_1$ and $M_2$ are not isomorphic.