Show that a semigroup $S$ is a rectangular band if and only if $ab=ba \Rightarrow a=b$

Question

Show that a semigroup $S$ is a rectangular band if and only if $ab=ba \Rightarrow a=b$. (For all $a,b\in S$)

I have the definition of a rectangular band as $aba=a$.

When I try to prove this I keep getting stuck. This is my best effort.

$aba=a$

$ab=ba \Rightarrow aba=baa \\ \Rightarrow a=baa \\ \Rightarrow ab=baab=b \\ \Rightarrow ab=b $

But I can't get from here to the final result. Any hellp is much appreciated

Answer 1

The opposite direction can be proved as follows. Since $a$ and $a^2$ commute, $a = a^2$. Now, since $(aba)a = aba^2 = aba = a^2ba = a(aba)$, $a$ and $aba$ commute and thus $aba = a$.

Answer 2

You're almost there. If we replace the variables to the opposite of what you have above, that is $$bab=a$$ $$ba=ab \Rightarrow bab=abb \\ \Rightarrow b=abb \\ \Rightarrow ba=abba=a \\ \Rightarrow ba=a$$ This is valid because $a$ and $b$ can be any elements of $S$. Therefore,$$b=ab=ba=a \\ \Rightarrow a=b$$

This proves the forward direction.