This question is related to and inspired by the question Why are groups more important than semigroups?.
I am curious why I don't see much studies on right groups.
On Pp.37 of Clifford& Preston's Algebraic Theory of Semigroups Vol. I : A semigroup is called a right group if it is right simple and left cancellative. It is equivalent to saying that for any element $a$ and $b$ of a semigroup $S$, there exists one and only one element $x$ of $S$ such that $ax = b$.
A right group is the direct product of a group and a right zero semigroup. A right group is the union of a set of disjoint groups. A periodic semigroup is a right group iff it is regular and left cancellative, etc.
A left group is the dual of a right group.
Most literature I found about right groups is very old (back in 50's and earlier). As far as I know, there is not much research on this subject in the past 30 years or so. Am I wrong about this? Or "right group" are common words so I do not get useful results when I google for it?
But I thought the right group concept could serve as a bridge between groups and semigroups. For example, a permutation group is a set of bijective mappings and a right zero transformation semigroup is a set of constant mappings, you have a right group if you take the direct product of a permutation group and a right zero transformation semigroup. What is this right group look like? Another example, an aperiodic semigroup contains no non-trivial groups, then what is a semigroup which contains no non-trivial right groups? What is a semigroup which contains a non-trivial right group? etc. etc.