The natural partial order on the idempotents of a semigroup is defined by $$e \leq f \; \; \text {iff} \;\; ef = fe = e$$ My question regards idempotents and Green's relations:
Can two different idempotents $e$ and $f$ that are $e\leq f$ be in the same $\mathcal D-$class?
From the finite examples I tried to build it does not seem to be the case, but maybe a more complex example is required.
There are no finite examples (easy to prove). In the bicyclic semigroup $\langle p, q\mid pq=1\rangle$ idempotents form sa chain and are D-related.