This is extracted from this question following Benjamin Steinberg's suggestion.
For a semigroup $S,$ let $P(S)$ denote the power semigroup of $S,$ which is made up of all non-empty subsets of $S$ with the operation $AB=\{ab\,|\,a\in A,b\in B\}.$
I'm thining about the following conditions $(P1)$ and $(P2)$ on $S:$
$(P1)$ Every left-cancellable element of $P(S)$ is a singleton.
$(P2)$ Every cancellable element of $P(S)$ is a singleton.
1) Are the two conditions equivalent? Are they equivalent for left-cancellative semigroups? Are they equivalent for cancellative semigroups?
2) Can we characterize semigroups satisfying these conditions? Left-cancellative ones? Cancellative ones?
The questions in bold interest me the most. The class of cancellative semigroups satisfying $(P1)$ contains groups and commutative semigroups. From above, any left-cancellative semigroup satisfying $(P1)$ will have to satisfy the right Ore condition.