A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely using multiplication, it also makes sense in the context of semigroups, and von Neumann regularity does make an appearance in that context as well. See this link for more information on the basic concept. An element $a\in R$ is strongly regular if $a\in a^2R\cap Ra^2$.
In recent joint work, I've been studying a property of elements in rings which is initially defined using addition, but has an equivalent completely multiplicative definition. Thus, I'm interested to know if it has been studied in the context of semigroups. It is defined, for an element $a$, by the property:
$$\exists b\ :\ aba=a,\ bab=b,\ \text{ and }\ b\ \text{ is strongly regular.}$$
For those semigroup-theorists out there, have you seen this condition (or anything like it) before? A related condition we also care about is the following:
$$\exists b\, :\ aba=a,\ bab=b,\ b^2=b.$$
