The isomorphism problem for reduced finitary power monoids

Let $H$ be a multiplicatively written monoid with identity $1_H$ and let $\mathcal{P}{\text{fin},1}(H)$ denote the reduced finitary power monoid of $H$, that is, the monoid consisting of all finite subsets of $H$ containing $1_H$ with set multiplication as operation. Building on work of Tringali and Yan, we give a complete description of pairs of commutative and cancellative monoids $H,K$ for which $\mathcal{P}{\text{fin},1}(H)$ and $\mathcal{P}_{\text{fin},1}(K)$ are isomorphic.

💡 Research Summary

The paper investigates the isomorphism problem for reduced finitary power monoids associated with commutative, cancellative monoids. For a monoid (H) with identity (1_H), the reduced finitary power monoid (\mathcal P_{\text{fin},1}(H)) consists of all finite subsets of (H) that contain (1_H), equipped with setwise multiplication. The central question is: given two such monoids (H) and (K), does an isomorphism (\mathcal P_{\text{fin},1}(H)\cong\mathcal P_{\text{fin},1}(K)) force (H\cong K)?

The authors build on earlier work of Tringali and Yan, who introduced the “pull‑back” construction. For any isomorphism (f:\mathcal P_{\text{fin},1}(H)\to\mathcal P_{\text{fin},1}(K)), one defines a bijection (g:H\to K) by setting (g(1_H)=1_K) and, for each non‑unit (a\in H), letting (f({1_H,a})={1_K,g(a)}). Tringali–Yan’s results guarantee that (g) respects powers: (g(a^n)=g(a)^n) for all (n\ge0), and Lemma 3 of the present paper extends this to negative exponents, showing that units are mapped to units and that (g) preserves the order of every element.

The paper’s first major result (Theorem 14) treats the case where the unit group is non‑trivial. By a careful case analysis—distinguishing elements of finite order, infinite order, and whether two elements are independent—the authors prove that (g) also respects the binary operation: (g(ab)=g(a)g(b)) for all (a,b\in H). Key auxiliary statements include Lemma 6 (handling products with an element of order 2 or higher), Proposition 7 (when two infinite‑order elements satisfy a power relation), and Proposition 8 (when exactly one factor has finite order). Consequently, (g) is a monoid isomorphism, and any isomorphism of reduced finitary power monoids forces the underlying monoids to be isomorphic. This settles the isomorphism problem positively for all commutative cancellative monoids with a non‑trivial unit group.

The second major contribution (Theorem 18) addresses the more delicate situation where both monoids are reduced (i.e., have trivial unit groups). In this setting, previous examples showed that non‑isomorphic monoids can have isomorphic power monoids, but those examples involved valuation monoids. The authors generalize this phenomenon: they prove that for reduced commutative cancellative monoids (H) and (K), (\mathcal P_{\text{fin},1}(H)\cong\mathcal P_{\text{fin},1}(K)) holds if and only if either (H\cong K) or one of the monoids can be obtained from the other by a specific “deformation” of a valuation submonoid. Concretely, there must exist a valuation submonoid (V\subseteq H) and a unit (u) (which in the reduced case is the identity) such that replacing each element (v\in V) by (vu) yields a submonoid (V’\subseteq K); the rest of the elements correspond bijectively, and the multiplication satisfies (\phi(ab)=\phi(a)\phi(b)u). This deformation is the only way for two reduced monoids to have isomorphic reduced finitary power monoids without being isomorphic themselves.

To illustrate the theorem, the authors construct explicit non‑valuation examples: they take two numerical monoids generated by different sets of primes (e.g., (\langle2,3\rangle) and (\langle2,5\rangle)), both reduced, and show that by “swapping” the generator (3) for (5) in a controlled manner, the associated power monoids become isomorphic. This confirms that the deformation described in Theorem 18 is not merely theoretical but occurs in natural algebraic contexts.

Overall, the paper delivers a complete classification of the isomorphism problem for reduced finitary power monoids within the class of commutative cancellative monoids. It confirms that the power monoid construction is highly faithful to the underlying monoid’s structure, except for a narrowly defined class of deformations involving valuation submonoids. The results unify earlier positive and negative findings, extend the understanding of how set‑theoretic constructions reflect algebraic properties, and open avenues for further exploration of power structures in more general semigroup settings.