Example of a cancellative power semigroup Let $ \, S \, $ be a semigroup such that $ \ |S| \geq 2 \ $. Its power semigroup is the power set $ \, \wp(S) \, $ together with the binary operation
$$XY = \{ xy \in S : x \in X, \ y \in Y \} \ \ .$$
I am interested in the semigroup $ \ Q_S = \wp(S) \setminus \{ \varnothing \} \ $, which I also call the power semigroup of $ \, S \, $.
A semigroup $ \, S \, $ is said left cancellative if, and only if, for all $ \ x,y,z \in S \ $, if $ \ xy=xz \ $, then $ \ y=z \ $.
A semigroup $ \, S \, $ is said right cancellative if, and only if, for all $ \ x,y,z \in S \ $, if $ \ yx=zx \ $, then $ \ y=z \ $.

I would like to see an example of a semigroup $ \, S \, $ such that $ \, Q_S \, $ is left cancellative and right cancellative.

I tested the most immediate and the most standard examples of semigroups, but none of them resulted in such a desired example. I don't know where to look anymore.

The set of positive integers under addition forms a cancellative semigroup, this is clearly not a group embeddable semigroup.
