Algebra of overpartitions

TL;DR

This paper extends the algebraic structure of partitions to overpartitions, showing they form an Abelian group and are isomorphic to positive rationals, advancing the algebraic understanding of overpartition theory.

Contribution

It introduces a multiplicative group structure for overpartitions and establishes their isomorphism with positive rationals, expanding algebraic frameworks in partition theory.

Findings

Overpartitions form an Abelian group under partition multiplication.

Overpartitions are isomorphic to positive rational numbers as groups.

Discussion of potential ring theory for overpartitions.

Abstract

In a 2022 paper, Dawsey, Just and the present author prove that the set of integer partitions, taken as a monoid under a partition multiplication operation I defined in my Ph.D. work, is isomorphic to the positive integers as a monoid under integer multiplication. In this note, I extend partition multiplication to the set of overpartitions, which are of much interest in partition theory. I prove the overpartitions form an Abelian group under partition multiplication. Moreover, the overpartitions and the positive rational numbers are isomorphic as multiplicative groups. I then prove further overpartition isomorphisms and discuss approaches to a ring theory of overpartitions.