This is not a great question for sure and it may even be trivial for all I know, but a couple of years ago, when I still thought I'd be a mathematician, I spent quite a lot of time thinking about it and my advisor wasn't able to help so I'd like to ask it here, even if only for the feeling that my effort wasn't completely wasted if anybody reads it. Hopefully that's OK.

Two semigroups are called globally isomorphic if their power semigroups are isomorphic. The power semigroup $P(S)$ of a semigroup $S$ is the semigroup of all non-empty subsets of $S$ with the natural multiplication.

Mogiljanskaja gave examples of semigroups which are globally isomorphic but not isomorphic, and more examples have been produced since. All of those examples, at least the ones I've seen, are semigroups with zero which differ on their annihilators, say, $\mathrm{Ann}(S) = \{s\in S\,|\,(\forall r\in S)\, rs=0\}$ or ones built from such semigroups. In general, there doesn't seem to be a great variety of examples. Mogiljanskaja asked what other kinds of examples could be found, and the general question seems to be: "what of the semigroup structure does a power isomorphism have to preserve?"

The first question that came to my head was if it was possible to have a semigroup with zero globally isomorphic to a semigroup without zero. Obviously, the power semigroup of a semigroup with zero has a zero as well, so we would have to have a semigroup without zero whose power semigroup has a zero. It's easy to construct such semigroups, and they have been studied. A semigroup whose power semigroup has a zero is called a homogroup and there's a paper by Clifford and Miller from 1948 about them, giving a general construction: *Semigroups Having Zeroid Elements*, A. H. Clifford and D. D. Miller, *American Journal of Mathematics*, Vol. 70, No. 1 (Jan., 1948), pp. 117-125. (JSTOR)

They are exactly the semigroups which have a two-sided ideal that's a group. (I know that thanks to this question.) That has to be the least ideal of the semigroup and so its kernel. Any semigroup with zero is a homogroup and $\{0\}$ is that ideal in those.

So the question is whether a semigroup without zero can be globally isomorphic to a homogroup with a bigger kernel. And another question that immediately comes to mind is whether, simply, two globally isomorphic homogroups can have non-isomorphic kernels.

So these are the questions I'd like to ask. If there are such pairs, I think that would be genuinely interesting. Probably there aren't though and the question is boring, but I wasn't able to prove it.

terribleterminology! $\endgroup$