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?