Let $[A,\circ]$ be a semi-group. Further $\forall a,b$ if $a \neq b$, then $a\circ b \neq b \circ a$. Now show that
$a\circ b\circ a=a$
My argument is as follows:
since in semi-group we have Associativity property,
So, we have $(a\circ b)\circ a=a \\ a\circ(b\circ a)=a$
So, seems like we have en identity element for this semi-group and that is $a\circ b=b\circ a$.
Hence, $a\circ b \circ a=a$
Am I correct in my argument?
Let $a \in A$ and let $b = a^2$. Then $ab = ba$ and hence $b = a$, that is, $a^2 = a$. Thus your semigroup is a band or, if you prefer, an idempotent semigroup. Now, consider $c = aba$. Then $ac = aaba = aba = c$ and $ca = abaa = aba = c$. Thus $ac = ca$, whence $a = c = aba$.