Could $ \mathcal{R}$-unipotent semigroups be defined as the regular semigroups in which the idempotents form a right normal band?

Question

A semigroup $S$ is said to be $\mathcal{R}-$unipotent if its idempotent form a left regular band, that is, $efe=ef$ for any $e,f$ idempotents in $S.$

A right normal band $E$ is a semigroup of idempotents in which $efg=feg$ for any $e,f,g$ in $E$.

Could $ \mathcal{R}$-unipotent semigroups be defined as the regular semigroups in which the idempotents form a right normal band?

Answer 1

Certainly not. If the two conditions were equivalent, then any $\mathcal{R}$-unipotent semigroup would satisfy, for all idempotents $e$, $f$ and $g$, $efe = ef$ and $efg = feg$. Taking $g = e$ would give $efe = fe$, whence $ef = fe$. Thus the idempotents of a $\mathcal{R}$-unipotent semigroup would commute, which is not the case (see the counterexample to your previous question).