Concerning inverse monoids

Question

Let $S$ be an inverse monoid and $y,u\in S$.

If $yu=1$, then $yuy=y$ and $uyu=u$. Are $u$ and $y$ inverse of each other?

Answer 1

In semigroup theory, an element $y$ is an inverse of $x$ if $x = xyx$ and $y = yxy$. Thus the answer to your question is yes. Note however, that $yu = 1$ does not imply $uy = 1$. Thus is $u$ and $y$ are not in general inverses in the group sense.