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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.

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    It might be helpful if you defined what a right group actually is in the question.2012-05-01
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    @ymar, I am not too sure about that. I forgot to mention being a right group is equivalent to saying the semigroup is right simple and contains an idempotent element. And the idempotents are the identities of the disjoint groups.2012-05-01
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    One more thing, every finite semigroup has an idempotent element. So, all finite right simple semigroups are right groups.2012-05-01
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    What do you mean by "What [does] this right group look like?" It looks like a direct product of a group and a right semigroup. What more do you exactly want to know?2012-05-02
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    @Tara B, I meant the right group transformation semigroup which is isomorphic to the direct product of a proper subgroup of $S_n$ and a right zero transformation semigroup which is of cardinality less than n. The rank of the elements of the premutation group is n. the rank of the elements of the right zero transformation semigroup is 1. What is this right group transformation semigroup? This is the kind of questions I had in mind. I don't have answers. That's one of the reasons I ask this question.2012-05-03
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    Actually I was hoping somebody can tell me I was asking a nonsense question such as why 1+1=2. Then I can drop this stupid thing and move on to other things. But, A group is a right group, a right zero semigroup is a right group, the direct product of two right groups is again a right group, a subsemigroup of a right group is a right group. Once I figure out these, how do I give up? I am documenting my findings here. Later on, I will convert these comments into a better question.2012-05-03

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