is $a \circ b \circ a=a$ when $[A,\circ]$ is a semi-group

Question

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?

How does $a\circ (b\circ a)=(a\circ b)\circ a$ imply that $a\circ b\circ a=a$? — Rushabh Mehta, Nov 10 '19 at 16:54
You assumed that $a \circ b \neq b \circ a$ and concluded that $a \circ b = b \circ a$ satisfies something, so this cannot be correct. — Levi, Nov 10 '19 at 17:14
So, how to proceed in this one.I am not getting any clue, other than what I posted, which came out to be wrong :/ — Aayush Upadhyaya, Nov 11 '19 at 04:37

score 1 · Answer 1 · answered Nov 10 '19 at 17:24

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 $a = c = aba$.

is $a \circ b \circ a=a$ when $[A,\circ]$ is a semi-group

1 Answers1