Let $R$ be a ring or a semigroup. We say that $x\in R$ is a von Neumann regular element of $R$ if there exists $y\in R$ such that
Any $y\in R$ satisfying the above equation is called a pseudoinverse of $x$. (It is usually not unique.) It is a useful notion which extends outside ring and semigroup theory. For example, it can be proven that for any $(m\times n)$-matrix $A$ over a field $K$ there exists (and can be effectively found) a $(n\times m)$-matrix $X$ over $K$ such that
This pseudoinverse of $A$ can be used to solve the linear system
This equation has solutions iff $Xb$ is a solution, which is easy to verify. It isn't much more difficult to check that if $Xb$ is a solution, then
where $y$ is arbitrary, gives all solutions $x$ of the system.
Let's say that $y$ is a 2-pseudoinverse of $x$ if
It is easy to see that if $x$ has a pseudoinverse $y$, then it has a 2-pseudoinverse too. Indeed, in this case, we have
so $yxy$ is a 2-pseudoinverse of $x.$
I would like to know whether the existence of a 2-pseudoinverse implies the existence of a pseudoinverse. That is, if I have $y$ such that
will I always find $y'$ such that
In yet other words, if $y$ is a regular element with a pseudoinverse $x,$ must $x$ be a regular element? (Is it true in semigroups? Is it true in rings?)
The problem is that nothing in the equation $yxy=y$ allows me to reduce some expression to $x,$ which is essentially what I would have to do in order to prove that the answer is "yes". On the other hand, to prove it is "no", I would have to find a counter-example and I have no idea where to look.
It is clear that I can't look for counter-examples in semigroups or rings in which all elements are regular. (Which are called regular semigroups and von Neumann regular rings respectively). Unfortunately, I know next to nothing about regular elements in semigroups which are not regular and in rings which are not von Neumann regular.