Power Semigroups and Two Rigidity Theorems for Groups

Technical Summary: Power Semigroups and Two Rigidity Theorems for Groups

Problem Statement
The paper addresses the isomorphism problem for power semigroups. Given a semigroup $H$, let $P(H)$ denote the "large power semigroup" (the set of all non-empty subsets of $H$ under setwise multiplication) and $P_{fin}(H)$ denote the "finitary power semigroup" (the set of all non-empty finite subsets of $H$). The central questions investigated are:

1. Global Rigidity: If $H$ belongs to a specific class of semigroups $\mathcal{O}$ and $P(H) \cong P(K)$ for some semigroup $K$, does it follow that $K \in \mathcal{O}$ and $H \cong K$?
2. Finitary Rigidity: Similarly, if $P_{fin}(H) \cong P_{fin}(K)$, does it follow that $K \in \mathcal{O}$ and $H \cong K$?

While it is known that the class of all groups is globally determined (i.e., $P(H) \cong P(K) \implies H \cong K$ for groups), the behavior of the finitary power semigroup $P_{fin}(H)$ for infinite groups is less understood. The paper specifically targets the rigidity of $P_{fin}(H)$ when $H$ is an additive subgroup of the rational numbers $\mathbb{Q}$.

Methodology
The authors employ a combination of structural semigroup theory, combinatorial number theory, and Diophantine geometry.

• Structural Analysis of Isomorphisms: In the infinitary setting (Section 2), the authors analyze global isomorphisms $f: P(H) \to P(K)$. They introduce the concept of "unit-stable" elements in power semigroups to prove that any global isomorphism between monoids restricts to an isomorphism between their respective unit groups. This structural result allows them to deduce that if $H$ is a group and $P(H) \cong P(K)$, then $K$ must also be a group.
• Combinatorial Connections: In the finitary setting (Sections 3 and 4), the proof relies on establishing a connection between the structure of $P_{fin}(H)$ and the Fibonacci sequence. Specifically, the authors prove that the number of solutions $X \in P_{fin}(H)$ to the equation $\{0, 1\} + X = \{0, 1, \dots, n\}$ (where $H$ contains an element of infinite order) is exactly the $n$-th Fibonacci number, $F_n$.
• Representation Theory and Diophantine Bounds: The authors define "indexed signed binary representations" (sums of the form $\sum 2^{e_i}\epsilon_i$). They utilize a deep result from Diophantine geometry—the Evertse–Schlickewei–Schmidt theorem on non-degenerate solutions to linear equations in multiplicative groups—to show that for any fixed length $k$, only finitely many Fibonacci numbers $F_n$ admit an indexed signed binary representation of length at most $k$.
• Reduction to Triviality: The proof of the main theorem proceeds by contradiction. Assuming $P_{fin}(H) \cong P_{fin}(K)$ where $H \le \mathbb{Q}$, the authors first establish a structural splitting $K \cong L \times H$, where $L$ is a submonoid with a trivial unit group. They then demonstrate that if $L$ were non-trivial, the size of certain sets in $P_{fin}(K)$ would be bounded, leading to a contradiction with the growth properties of Fibonacci numbers and the finiteness results derived from the Evertse–Schlickewei–Schmidt theorem.

Key Contributions and Results

1. Theorem 1.3 (Infinitary Rigidity): The authors provide a proof that if $H$ is a group and $K$ is a semigroup such that $P(H) \cong P(K)$, then $H \cong K$. This is established as a corollary of a more general result (Theorem 2.3): every global isomorphism from a monoid $H$ to a monoid $K$ restricts to a global isomorphism from the unit group $H^\times$ to $K^\times$.
2. Theorem 1.4 (Finitary Rigidity for Subgroups of $\mathbb{Q}$): The paper proves that if $H$ is an additive subgroup of the rational numbers $\mathbb{Q}$ and $K$ is a semigroup such that $P_{fin}(H) \cong P_{fin}(K)$, then $H \cong K$. This resolves the finitary rigidity problem for the class of groups isomorphic to subgroups of $\mathbb{Q}$ (which includes $\mathbb{Z}$, $\mathbb{Q}$, and groups like $\mathbb{Z}[1/p]$).
3. Combinatorial Lemma (Proposition 3.2): The paper establishes a novel link between power semigroups and the Fibonacci sequence, showing that the number of solutions to a specific setwise addition equation in a monoid with an element of infinite order is given by Fibonacci numbers.
4. Finiteness of Representations (Lemma 3.4): The authors prove that for any fixed $k$, there are only finitely many $n$ such that the Fibonacci number $F_n$ has an indexed signed binary representation of length at most $k$. This result is derived using the Evertse–Schlickewei–Schmidt theorem.

Significance and Claims
The paper claims to settle the finitary isomorphism problem for a fundamental class of infinite groups (subgroups of $\mathbb{Q}$), a case noted as "considerably more difficult" than the infinitary case. The significance lies in the methodology: the proof of the finitary rigidity theorem relies, in a "circuitous way," on a special case of the Evertse–Schlickewei–Schmidt theorem, a result from the theory of linear equations in multiplicative groups. This connects the structural theory of power semigroups with deep results in Diophantine geometry.

The authors explicitly state that the proof for the finitary case cannot be derived from the infinitary results because the unit group of an infinite monoid need not belong to the finitary power semigroup, preventing the direct application of the unit-stability arguments used in Section 2.

Future Directions
The paper concludes by conjecturing that Question 1.2(2) (finitary rigidity) has a positive answer for the class of all groups. The authors suggest that the next logical step is to settle the question for torsion-free abelian groups, noting that their current result covers the specific case of subgroups of $\mathbb{Q}$. They also pose an open question regarding whether an isomorphism between finitary power semigroups of monoids must restrict to an isomorphism between the finitary power semigroups of their unit groups.