Inverse of a product in semigroup Disclaimer. The following might end up being a stupid question
Let $S$ be a regular semigroup i.e for every $s\in S$ we can write $s=sas$ for some $a\in S$ (called an inverse of $s$).

Do we know how to compute explicitly an inverse of a product? That is given $s=sas$ and $t=tbt$, can we tell what an inverse of $st$ would be?

Some attempts go as follows
$$st(btsa)st = s(tbt)(sas)t = sstt =st \Leftarrow s,t\in E(S) $$
We could also do
$$ st(bca)st = (sas)t(bca)s(tbt) = s(astb)c(astb)t = s(astb)t = (sas)(tbt) = st $$
We've assumed $c\in V(astb)$, which is provided by regularity, and the arithmetic checks out, but I'm not entirely convinced by this. 
 A: You can't find such an explicit inverse in general and essentially because of what has been said already by M. Vinay on 24.03, the first comment. Also, consider the simpler case with inverse semigroups. There the inverse of a product is very simple but in a regular semigroup, there is no the inverse but the set of inverses and this leaves everything open, at least up to the precision of that set of inverses $V(s)$ for any $s\in S$.
More precisely, I think the originial question is equivalent of asking: given any $s\in S$, give explicitly the inverse of $s$.
Proof.
One way is obvious: if one can give an explicit inverse for any $s\in S$ then this also holds for $st\in S$ for any $s,t\in S$.
The reverse implication: if one can give explictly an inverse of $st$ for any $s,t\in S$ then just note that if $s\in S$ is arbitrary and $s'\in V(s)$ then $s=ss's=s(s's)$ and explicit inverse of this can be given by assumption. QED.
I would also like to give something that may give you hints in this direction. In Howie's "Fundamentals of Semigroup Theory" (1995), there is Theorem 2.5.4 on page 60:
Let $a,b\in S$, where $S$ is a regular semigroup. Let $a'\in V(a)$, $b'\in V(b)$ and $g\in S(a'a,bb')$. Then $b'ga'\in V(ab)$.
Here $S(e,f)$ (for any two idempotents $e,f\in E(S)$) is the sandwich set defined as follows: $S(e,f)=\{g\in V(ef)\cap E:ge=fg=g\}$ and has an alternative description $S(e,f)=\{g\in E:ge=fg=g, egf=ef\}$.
So considering the original question, this $b'ga'$ is one example of an inverse of $ab\in S$ and one can try to improve this fundamental result or prove that this can't be improved any further, or can show similar connection using some other set than the sandwich set.
One more comment. Considering any regular semigroup, one can use Nambooripad's general description of regular semigroups but I'm not sure where this really leads to or if it gives any useful information.
