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The isomorphism problem for reduced finitary power monoids

Balint Rago

TL;DR

This work resolves the isomorphism problem for reduced finitary power monoids $\mathcal{P}_{\mathrm{fin},1}(H)$ in the setting of commutative cancellative monoids. Building on the pullback construction from prior work, it split the analysis into non-reduced and reduced cases, proving a positive isomorphism result when a nontrivial unit group is present. For the reduced case, it introduces a canonical decomposition via pseudo-units, showing that a reduced monoid $K$ is essentially determined by a valuation submonoid and a complementary part, leading to a full classification: $\mathcal{P}_{\mathrm{fin},1}(H)\simeq\mathcal{P}_{\mathrm{fin},1}(K)$ if and only if $H\simeq K$ or both are reduced and $K$ is built from $H_N$ and a reduced valuation monoid with matching quotient groups. The results generalize previous work and include explicit non-valuation examples, clarifying the role of deformation around valuation submonoids in the isomorphism problem.

Abstract

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.

The isomorphism problem for reduced finitary power monoids

TL;DR

This work resolves the isomorphism problem for reduced finitary power monoids in the setting of commutative cancellative monoids. Building on the pullback construction from prior work, it split the analysis into non-reduced and reduced cases, proving a positive isomorphism result when a nontrivial unit group is present. For the reduced case, it introduces a canonical decomposition via pseudo-units, showing that a reduced monoid is essentially determined by a valuation submonoid and a complementary part, leading to a full classification: if and only if or both are reduced and is built from and a reduced valuation monoid with matching quotient groups. The results generalize previous work and include explicit non-valuation examples, clarifying the role of deformation around valuation submonoids in the isomorphism problem.

Abstract

Let be a multiplicatively written monoid with identity and let denote the reduced finitary power monoid of , that is, the monoid consisting of all finite subsets of containing with set multiplication as operation. Building on work of Tringali and Yan, we give a complete description of pairs of commutative and cancellative monoids for which and are isomorphic.
Paper Structure (4 sections, 18 theorems, 117 equations)

This paper contains 4 sections, 18 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the pullback of an isomorphism
  3. Is the pullback an isomorphism?
  4. The case of reduced monoids

Key Result

Theorem 1

Let $H,K$ be monoids and $f:\mathcal{P}_{\mathop{\mathrm{fin}}\nolimits,1}(H)\to\mathcal{P}_{\mathop{\mathrm{fin}}\nolimits,1}(K)$ an isomorphism. Then $f(X)$ is a 2-element set for every 2-element set $X\in\mathcal{P}_{\mathop{\mathrm{fin}}\nolimits,1}(H)$.

Theorems & Definitions (35)

  • Theorem 1: 3Tr-Ya25
  • Proposition 2: 3Tr-Ya25
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 25 more