Let $M$ be a monoid, and let $x$ and $y$ be non-invertible elements of $M$. Must the product $x*y$ also be non-invertible? I am especially interested in the case where neither $x$ nor $y$ have neither a right nor a left inverse.
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3Does this answer your question? Can the product of two non invertible elements in a ring be invertible? – Anne Bauval Sep 20 '22 at 06:19
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No. Take $M$ to be the monoid of functions $f : \mathbb{N} \to \mathbb{N}$ under composition, take $y$ to be the function $y(k) = k + 1$, and take $x$ to be the function $x(k) = \text{min}(0, k-1)$. Then $y$ is not surjective and $x$ is not injective but their composition is the identity.
If $xy$ is invertible then $x$ necessarily has a right inverse and $y$ necessarily has a left inverse, so if we assume that either of those inverses doesn't exist then by taking the contrapositive, $xy$ is non-invertible.
However, this is true if $M$ is finite, and this follows from the fact that left inverses and right inverses are inverses in this case.
Qiaochu Yuan
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