Is the class of inverse semigroups globally determined?

Question

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.

Answer 1

(comment) I do not know the answer to your question, but I do know the reason why sometimes it is convenient to remove the empty sets. This is because more properties of the semigroup are preserved in passing to its power. For example, if S is a rectangular band (that is, it satisfies xyz=xz) then $2^S \setminus \{ \emptyset\}$ is still a rectangular band, while $2^S$ is not (take $y= \emptyset$ and $x, z$ nonempty).

There are general results characterizing what is preserved in passing from some structure to its power, even in a general algebraic setting (structures with any number and any kind of operations, possibly also relations, even infinitary). If you disallow the empty set, satisfaction of linear equation is preserved ("linear" means that each variable appears at most once on each side). If you allow the empty set, you have to deal with balanced linear equations (this means that each variable appears exactly once on each side). For example, the above equation xyz=xz is linear but not balanced.

Probably, such arguments are not of much use in the case of inverse semigroups, since I do not see how to define them only in terms of linear equations (even allowing inverse as a unary operation). However, in case someone is interested, references for the general cases can be found in Brink, C.: Power structures. Algebra Universalis 30, 177–216 (1993) or Gratzer G., Whitney S., Infinitary varieties of structures closed under the formation of complex structures, Colloq. Math. 48 (1984) 1-5.

Answer 2

This is not a proof yet, but reduces the question to a simpler one. I think that inverse semigroups are not determined up to isomorphism by their power semigroups. Indeed, consider any countable linear order $L$. It can be viewed as a semilatice (=idempotent inverse semigroup) $S(L)$ with $ab=a$ iff $a\le b$. Order $L$ is isomorphic to $L'$ iff the semigroups $S(L), S(L')$ are isomorphic. Every subset $S$ of $L$ satisfies $SS=S$. So $2^L$ is a semilattice too. The product of two subsets $XY$ consists of all elements of $X$ which are not bigger than some element of $Y$ and all elements of $Y$ which are not bigger than some element of $X$. In other words, let $a$ be the supremum of elements of $X$, $b$ be the supremum of elements of $Y$. Assume that $a\ge b$. Then $XY$ consists of all elements of $X\cup Y$ which do not exceed $b$.

Now let $L$ be the order $\mathbb{Q}$ and $L'$ be $L$ without 0. Are $2^L$ and $2^{L'}$ isomorphic? My conjecture is that the answer is ``yes" and this is an example of two non-isomorphic inverse semigroups with isomorphic power semigroups. A possibly easier example would be $L=\{-1/n, 0, 1/n\mid n=1,2,...\}$, $L'=\{1/n,-1/n\mid n=1,2,...\}$.

Update. No, for linear orders, the rigidity is true (if $2^L=2^{L'}$, then $L=L'$). Indeed, first the element $L\in 2^L$ can be described as the only element $x$ such that $\forall y (xy=x \& \forall t (yt=t\to xt=t) ) \to y=x$. Then let $U=L2^L$, i.e. the ideal generated by $L$ in $2^L$. It consists of all ideals of the order $L$. An element $x\in U$ has the maximal element in $L$ if and only if there exists $m=m(x)\in 2^L$ such that

• $Lm=x$, and

• for every $n\in 2^L$ such that $Ln=x$ we have $an=n$.

Now 1-element subsets in $2^L$ are those $m\in 2^L$ for which there exists $x\in U$ such that $m=m(x)$.

The order on 1-element subsets is defined as $m\le m'$ iff $mm'=m$. Thus if $2^L$ and $2^{L'}$ are isomorphic, then $L$ and $L'$ are isomorphic (since $L$ is elementary determined inside $2^L$).

Answer 3

[Not an answer, but too long for a comment.]

In the comments to the OP, Benjamin Steinberg noted that "there is a paper without proofs in a 1987 conference proceedings by Tamura" (excerpt from this comment, emphasis mine) where it is claimed that "all finite semigroups are globally determined" (excerpt from this comment). The paper alluded to by Benjamin is

T. Tamura, "On the recent results in the study of power semigroups", pp. 191–200 in: S.M. Goberstein and P.M. Higgins (eds.), Semigroups and their applications, Reidel Publishing Company, 1987.

In fact, I've still recently seen a couple of papers claiming, by a reference to Tamura's work, that the isomorphism problem for power semigroups is completely solved in the finite case. This, however, is plainly false. Let me quote from p. 5 of [H.B. Hamilton and T.E. Nordahl, Tribute for Takayuki Tamura on his 90th birthday, Semigroup Forum 79 (2009), 2-14]:

Tamura [...] was an initiator of the study of power semigroups of a semigroup, testing to see if $S$ is isomorphic to $T$ when $P(S)$ is isomorphic to $P(T)$. This question was answered in the negative for infinite semigroups by Mogiljanskaja, but Tamura was determined to show it had an affirmative answer for finite semigroups. One of the current paper's authors was still attending Tamura's Tuesday afternoon seminars during the 1980's and Tamura would show up most weeks with a new typed copy of his recent work on power semigroups. We would go over this new work with him and the complexity of it all was overwhelming to many of us. [...] This author had a conversation with Peter Jones at a conference on semigroups at which Peter told him that Tamura was sending him copy after copy of this power semigroup stuff and asked whether he should be reading it all or should wait until Tamura had the final results. The isomorphism problem for finite semigroups remains open.