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.

rightcancellation law on the elements of $S,$ i.e. that the multiplication by $x$ on the right side is injective in $S$. But more importantly, what $SX=SS$ says is not that $X$ is not right-cancellable, but that $S$ is not left-cancellable, which has no bearing on the question I'm asking since that not the same side that the cancellative property that holds in $S$. $\endgroup$ – Michał Masny Feb 3 '15 at 17:19