This is Exercise 1.9.4 of Howie's "Fundamentals of Semigroup Theory".
Definition: A semigroup $S$ is a rectangular band if $aba=a$ for all $a,b$ in $S$.
Lemma: Let $S$ be a rectangular band. Then every element of $S$ is an idempotent.
Proof: Let $a$ be an element of a rectangular band $S$. Then $a^3=a$ implies $a^4=a^2$. But $a^4=a(a^2)a=a$. Hence $a^2=a$.$\square$
The Question:
Show that a semigroup $S$ is $(A)$ a rectangular band if and only if $(B)$ for all $a,b$ in $S$, $ab=ba$ implies $a=b$.
My Attempt:
For $(A)\implies (B)$: Let $S$ be a rectangular band. Let $a, b\in S$ such that $ab=ba$. Then $$\begin{align} a&=aba \\ &= (ba) a \\ &=ba \\ &=b(ba) \\ &=bab \\ &=b. \end{align}$$
But I don't know how to show $(B)\implies(A)$.
Please help :)
Elements $a$ and $a^2$ commute, so by $(B)$ we have $a^2 = a$ (every element is an idempotent).
We infer that elements $a$ and $aba$ commute: $$ abaa = aba = aaba, $$ hence $(B)$ implies $aba = a$.