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Let $a, b$ be elements in a monoid such that $ab$ has an inverse. Is it true that $a$ and $b$ have inverses? Prove this if true or give a counterexample if false.

I believe this is false because let $c$ be the inverse of $ab$. Then we have $abc = cab = e$. So we have $a(bc) = (ca)b = e$. So $a$ has a right inverse and $b$ has a left inverse. However, this does not imply that $a$ also has a left inverse and $b$ also has a right inverse because it is not necessarily an abelian monoid. I cannot find a counterexample though. Can anyone help?

bob ispro
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    Do you know any examples of monoids with elements that have left inverses but not right inverses? Those would be the natural place to look for a counterexample. – Eric Wofsey Oct 19 '21 at 16:24

1 Answers1

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First of all, your definition of an inverse is wrong. In semigroup theory, an element $x$ is an inverse of an element $a$ if $axa = a$ and $xax = x$. Note that an element may have no inverse, but may also have several inverses.

That being said, the answer to your question is no. Take the four-element monoid $M = \{1, a, b, 0\}$, in which $1$ is the identity, $0$ is a zero and $aa = ab = ba = bb = 0$. Then $0$ is its own inverse, $0 = ab$, but neither $a$ nor $b$ have inverses.

J.-E. Pin
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