This question is a follow-up to this one I asked on math.stackexchange. I've decided to ask here because I believe this is a research-level question. I'm sorry if I'm wrong -- I'm not a researcher myself and may well be mistaken about it.
I will reitarate the definitions I posted in an answer to that question. For a semigroup $S,$ the set $\mathcal P (S)=2^S\setminus \lbrace \emptyset\rbrace$ with mutliplication given by $$AB=\lbrace ab\ |\ a\in A,b\in B\rbrace$$ is called the power semigroup of $S$. (I don't really know what removing the empty set from it makes any better, but this is what I found in most papers.) We say that two semigroups are globally isomorphic iff their power semigroups are isomorphic. We say that a class of semigroups $\mathscr C$ is globally determined iff any two globally isomorphic semigroups in $\mathscr C$ are automatically isomorphic, ie. for any $S,T\in\mathscr C$ we have $$\mathcal P(S)\cong\mathcal P(T)\implies S\cong T$$
The question I've linked to was about the class of all groups. As Steve D kindly pointed out in his answer, it is actually very simple to see that the class of all groups is globally determined. I started wondering if we can generalize this fact to all inverse semigroups, but in vain. There is little literature on this, even if I said otherwise previously. (I then meant that there was more than I'd expected.) Also, it seems to me that about 1990 papers on these things stopped appearing completely. Some that I've found are referenced on the linked page.