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There is a basic result about commutative semigroups which says

Any commutative semigroup satisfying the cancellation property can be embedded in a group.

However, I read that

Not every semigroup with cancellation can be embedded in a group.

Can someone provide a reference or an example of a (non-commutative) semigroup with cancellation that cannot be embedded in a group?

What else is known about algebraic structures with the cancellation law which may not be subsets of a group?

Caleb Stanford
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Sushil
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    https://en.wikipedia.org/wiki/Cancellative_semigroup#Embeddability_in_groups – Alastair Litterick Jan 22 '15 at 21:11
  • I have edited to improve your question. The wording was a little unclear; I also took the liberty of adding some details. If you can, please check if what I wrote is what you wanted. – Caleb Stanford Jan 24 '15 at 01:35
  • @Goos thanks and what you added I get to know that there is complete different study of algebric structure with cancellation law(This is much more than what I wanted) – Sushil Jan 26 '15 at 03:54

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