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On the automorphisms of the power semigroups of a numerical semigroup

This paper proves that the automorphism groups of both the power semigroup $\mathcal{P}(H)$ and the submonoid $\mathcal{P}_0(H)$ of a numerical semigroup $H$ are trivial, demonstrating that the only automorphism for these structures is the identity map.

Original authors: Salvatore Tringali, Kerou Wen

Published 2026-04-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Salvatore Tringali, Kerou Wen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a special box of numbers called a Numerical Semigroup. Think of this box as a club that only lets in non-negative whole numbers (0, 1, 2, 3...) and has a very strict rule: if you pick any two members from the club and add them together, the result must also be a member of the club. For example, if 3 and 5 are in the club, then 8 must be too.

Now, imagine you don't just play with single numbers, but with groups of numbers (subsets) from this club. Let's call this new, bigger box the Power Semigroup. In this new box, the "game" is to take two groups of numbers, mix them all together by adding every number in the first group to every number in the second group, and see what new group you get.

The big question the authors, Salvatore Tringali and Kerou Wen, asked is: How much can you rearrange the rules of this new game without changing the game itself?

In math terms, they are looking for automorphisms. Think of an automorphism as a "magic shuffle." If you have a deck of cards, a shuffle that keeps the deck looking exactly the same (even though you moved the cards around) is a trivial shuffle. But if you can shuffle the deck in a way that changes the order but the rules of the game (how cards combine) stay perfectly valid, that's a non-trivial automorphism.

The Main Discovery: The "Rigid" Box

The authors discovered something surprising and rigid about these number clubs:

The "Single Number" Club (Numerical Monoid): If your club includes the number 0, and you look at all the groups of numbers that must contain 0, the only way to shuffle them is to do absolutely nothing. The "magic shuffle" is just standing still. The group of all possible shuffles is trivial (it has only one member: the identity).
The "Any Group" Club (Numerical Semigroup): Even if you drop the rule that groups must contain 0, and just look at any non-empty group of numbers from the club, the result is the same. The only way to rearrange the groups while keeping the addition rules valid is to leave everything exactly where it is.

The Analogy of the Lock:
Imagine the Power Semigroup is a complex, high-tech lock made of many tumblers (the groups of numbers). Usually, you might think there are many different ways to turn the tumblers to open the lock (many different shuffles). However, the authors proved that for these specific number clubs, the lock is so perfectly engineered that there is only one key that fits: the key that doesn't turn at all.

How They Proved It (The Detective Work)

The authors didn't just guess; they used a mix of detective work from two different fields: Combinatorics (counting and arranging) and Semigroup Theory (the study of these number rules).

The Finite vs. Infinite Clue: They first showed that if you have a "magic shuffle" that works on the whole box (including infinite groups), it must also work perfectly on the smaller box containing only finite groups. It's like saying if you can rearrange a whole library without breaking the books, you can definitely rearrange just the books on one shelf.
The "Smallest Number" Trick: They found a way to identify the "smallest" number in any group. They proved that no matter how you shuffle the groups, the smallest number in a group must stay the same. It's like saying if you have a bag of marbles, and you shuffle the bags around, the lightest marble in the bag stays the lightest.
The "Identity" Connection: By proving that the smallest numbers can't move, and that the "shuffles" of the smaller groups are already known to be rigid, they showed that the big groups can't move either. If the foundation (the smallest numbers) doesn't budge, the whole building stands still.

Why This Matters (In the Context of the Paper)

The paper doesn't claim this will fix your car or cure a disease. Instead, it solves a specific puzzle in the world of pure mathematics.

Rigidity: It shows that these mathematical structures are incredibly "stiff" or "rigid." They don't have hidden symmetries or secret ways to rearrange themselves.
New Territory: Before this paper, mathematicians mostly studied how to shuffle finite groups of numbers. This paper is the first to successfully tackle the much harder problem of shuffling groups that can be infinite.
Future Hints: The authors suggest that this "rigidity" might be a common trait in other types of number systems, not just these specific ones. They propose a few guesses (conjectures) about whether this "no-shuffle" rule applies to other mathematical structures, like the set of all integers.

In a nutshell: The authors proved that for these specific number clubs, the rules are so strict that the only way to rearrange the groups of numbers without breaking the rules is to not rearrange them at all. The system is perfectly rigid.

1. Problem Statement

The paper investigates the structure of the automorphism groups of power semigroups derived from numerical semigroups.

Context: Let $H$ be a numerical semigroup (a cofinite subsemigroup of the non-negative integers $\mathbb{N}$ under addition). The power semigroup $P(H)$ consists of all non-empty subsets of $H$ equipped with the sumset operation $X + Y = \{x + y : x \in X, y \in Y\}$ .
Specific Structures: The authors focus on two specific structures:
1. $P(H)$ : The large power semigroup (involving both finite and infinite subsets).
2. $P_0(H)$ : The reduced large power monoid (submonoid of $P(H)$ containing subsets that include the identity element $0$).
Core Question: What is the automorphism group $\text{Aut}(P(H))$ and $\text{Aut}(P_0(H))$ ? Specifically, do these structures possess non-trivial automorphisms, or are they "rigid" (i.e., do they only admit the identity map)?

2. Methodology

The authors employ a blend of combinatorial arguments and semigroup theory. Their approach relies on several key methodological pillars:

Reduction from Infinitary to Finitary: A central challenge is that $P(H)$ $P (H)$ contains infinite sets, making standard combinatorial arguments difficult. The authors prove that any isomorphism between large power structures restricts to an isomorphism between their finitary counterparts (subsets of finite sets).
- Theorem 2.3 & 2.5: They demonstrate that for cancellative monoids/semigroups, an isomorphism $f: P_1(H) \to P_1(K)$ (or $P(H) \to P(K)$ ) necessarily maps finite sets to finite sets. This allows them to leverage existing results on finitary power monoids.
Algebraic Characterization of Set Properties: They convert set-theoretic properties (like element membership) into algebraic equations within the power semigroup.
- Lemma 3.2: They establish a condition where $y \in X$ if and only if $H_y + X = H^*_y + X$ , where $H_y$ and $H^*_y$ are specific idempotent sets. This allows the authors to test membership algebraically.
Idempotent Fixing: Using the fact that automorphisms preserve idempotents, they show that if an automorphism fixes certain generating sets (like singletons or sets $\{0, x\}$ ), it must fix all idempotents, and consequently, all subsets.
Conjugacy Quotients: For the general case of $P(H)$ (where $0$ might not be in the set), they utilize the concept of $T$ -conjugacy (where $T$ is the set of cancellative elements). They construct an isomorphism between the quotient semigroup $P(H)/\sim_T$ and the reduced power monoid $P_0(\mathbb{N})$ , effectively reducing the problem to a known case.

3. Key Contributions

The paper makes three primary theoretical contributions:

Restriction Theorems (Section 2):
- Proved that isomorphisms between reduced large power monoids of cancellative monoids restrict to isomorphisms of their finitary submonoids (Theorem 2.3).
- Proved that isomorphisms between large power semigroups of cancellative commutative semigroups restrict to isomorphisms of their finitary subsemigroups (Theorem 2.5).
- Significance: This bridges the gap between the well-studied finitary case and the more complex infinitary case, allowing the transfer of rigidity results.
Rigidity of Reduced Power Monoids (Section 3):
- Established that for any numerical monoid $H$ , the automorphism group of the reduced large power monoid $P_0(H)$ is trivial (Theorem 3.3).
- Mechanism: They showed that any automorphism must fix sets of the form $\{0, x\}$ (via Lemma 3.1) and subsequently fix all idempotents. Using the algebraic characterization of membership, they proved the automorphism must be the identity on all subsets.
Rigidity of General Power Semigroups (Section 4):
- Established that for any numerical semigroup $H$ , the automorphism group of the large power semigroup $P(H)$ is trivial (Theorem 4.3).
- Mechanism: By utilizing the $T$ -conjugacy relation, they mapped the problem to $P_0(\mathbb{N})$ . Since the automorphism group of $P_0(\mathbb{N})$ is trivial, the automorphism on the quotient is trivial. Combined with the fact that automorphisms preserve the minimum element of a set (Lemma 4.2), they proved the original automorphism must be the identity.

4. Main Results

Theorem 3.3: $\text{Aut}(P_0(H))$ is the trivial group for every numerical monoid $H$ .
Theorem 4.3: $\text{Aut}(P(H))$ is the trivial group for every numerical semigroup $H$ .

These results stand in contrast to some other algebraic structures where power semigroups have large automorphism groups (e.g., the automorphism group of the finitary power monoid of $\mathbb{Z}$ is isomorphic to $C_2 \times D_\infty$ ). The authors show that numerical semigroups exhibit a unique "rigidity" where the power structure admits no non-trivial symmetries.

5. Significance and Future Directions

Novelty: This is the first investigation into the automorphisms of infinitary power semigroups (involving infinite sets). Previous work was largely confined to finitary power semigroups.
Methodological Impact: The techniques developed, particularly the reduction from infinitary to finitary structures and the use of conjugacy quotients, provide a toolkit for analyzing power semigroups of other algebraic structures.
Conjectures: The authors propose two conjectures for future research:
1. Conjecture 5.1: The automorphisms of the large power monoid of the integers $\mathbb{Z}$ are all "inner" (induced by automorphisms of $\mathbb{Z}$ itself).
2. Conjecture 5.2: For any cancellative monoid $H$ , every automorphism of $P(H)$ maps the submonoid of sets containing the identity ( $P_1(H)$ ) to itself.
Broader Context: These results deepen the understanding of the relationship between a semigroup and its power semigroup, suggesting that numerical semigroups are structurally "rigid" in a way that prevents the power semigroup from acquiring new symmetries. This has implications for factorization theory and arithmetic combinatorics, where power semigroups are frequently used as models.