An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$

An element $a$ of a monoid $M$ is invertible iff there exists $x\in M$ such that $axa=1$

I can't do this one. How do I get started? It looks like it is saying there is only an inverse if $x=a^{-1}a^{-1}$ is in $M$, e.g. it is only invertible if there is an $x$ that is a left and right inverse of $a$, which makes sense, but then isn't the answer 'true by definition'?

• The problem isn't very hard, but you're tricking yourself with the notation $a^{-1}$, which you are not allowed to use until you have shown that $a$ does in fact have a two-sided inverse. But certainly, if $a$ is invertible and you put $x=a^{-1}a^{-1}$, then the given formula holds. So now you need only prove the converse. – Harald Hanche-Olsen Jan 16 '15 at 15:16
• @HaraldHanche-Olsen I just did $axa=1\implies ax=a^{-1}\implies x=a^{-1}a^{-1}$. You are saying I just need to say now $x=a^{-1}a^{-1}\implies ax=a^{-1}\implies axa=1$? – Makoto K. Jan 16 '15 at 15:18
• For the opposite take $x=(a^{-1})^2$. – Janko Bracic Jan 16 '15 at 15:20
• Don't forget that $a$ can have a left inverse and no right inverse, or vice versa. Do you know a result about the case when both a left inverse and a right inverse exist? – Harald Hanche-Olsen Jan 16 '15 at 15:20
• @HaraldHanche-Olsen I don't think so, actually I haven't read anything other than a passing comment on my previous question about left and right inverses – Makoto K. Jan 16 '15 at 15:21

The only hard part is proving that the existence of such an $x$ implies invertibility of $a$.

You have that $ax = ax(axa) = (axa)xa = xa$, so $a$ and $x$ commute.

Now you just need to conclude that $ax$ (which is equal to $xa$) is the unique inverse of $a$.

• Oh wow that is a novel approach! I wouldn't forget that! Did you think of that?? – Makoto K. Jan 16 '15 at 15:51
• Novel? It's a completely standard "trick" to write an expression and use the associative law to put parentheses two different ways to show two things are equal. – kahen Jan 16 '15 at 15:52
• Novel having not seen it before. Everything is novel the second before it becomes trivial(it was certainly novel when it was made aswell!) – Makoto K. Jan 16 '15 at 15:54
• See also this question. – Harald Hanche-Olsen Jan 16 '15 at 16:37

$(\Longrightarrow)$If $a$ is invertible, then $$az=za=1\implies \exists x,axa=1$$ Take $x=zz$, then $az=za=1 \implies (az)(za)=(1)(1)=1$

$(\Longleftarrow)$ There is some $x$ such that $axa=1 \implies az=za=1$

Proof: $$axa=1\implies ax=a^{-1}\implies (ax)a=a^{-1}a=1$$ So $a$ has a left inverse, similarly $$axa=1\implies xa=a^{-1}\implies a(xa)=aa^{-1}=1$$ so $a$ has a right inverse. It is know that the left and right inverse are equal(if both exist), so $ax=xa=a^{-1}=z$

Therefore $az=aa^{-1}=za=a^{-1}a=1$

$\blacksquare$

• "$x = a^{-1}a^{-1}$" has no meaning when you haven't proved that $a$ is invertible yet. – kahen Jan 16 '15 at 15:44
• Your $\Leftarrow$ calculation actually belongs in the $\Rightarrow$ section. What you have done is assume that $a$ is invertible and then verified by calculation that $axa = 1$ (making use of the fact that $a$ is left- and right-invertible). So far there is no justification for $\Leftarrow$ at all here :). – Erick Wong Jan 16 '15 at 15:51
• @ErickWong Can you fix my arrows haha, I can't understand the last part of your post because of the first part – Makoto K. Jan 16 '15 at 15:53
• @MakotoK. It might be better not fix this for you because it is an error of understanding and not a simple typo :). Can you see why the first chain of implications doesn't prove by itself that $axa = 1$? – Erick Wong Jan 16 '15 at 15:57
• Looks pretty good now, thanks! – Erick Wong Jan 16 '15 at 17:57