On semigroup algebras with rational exponents

TL;DR

This paper investigates the structure of Puiseux algebras, a class of semigroup algebras with rational exponents, establishing isomorphism conditions, atomic properties, and closure descriptions, revealing their complex factorization behaviors.

Contribution

It proves the isomorphism problem for Puiseux algebras, constructs classes with various atomic properties, and characterizes their closure structures, advancing understanding of their algebraic and factorization properties.

Findings

Isomorphism of Puiseux algebras implies monoid isomorphism.

Constructed Puiseux algebras with ACCP, bounded, and finite factorization properties.

Described seminormal, root, and integral closures, including antimatter cases.

Abstract

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in a field $F$ and exponents in an additive submonoid $M$ of $\mathbb{Q}_{\ge 0}$ is called a Puiseux algebra and denoted by $F[M]$. Here we study the atomic structure of Puiseux algebras. To begin with, we answer the Isomorphism Problem for the class of Puiseux algebras, that is, we show that for a field $F$ if two Puiseux algebras $F[M_1]$ and $F[M_2]$ are isomorphic, then the monoids $M_1$ and $M_2$ must be isomorphic. Then we construct three classes of Puiseux algebras satisfying the following well-known atomic properties: the ACCP property, the bounded factorization property, and the finite factorization property. We show that there are bounded factorization Puiseux algebras with extremal systems of sets of lengths, which allows us to prove that Puiseux algebras cannot be determined up to isomorphism by their arithmetic of lengths. Finally, we give a full description of the seminormal closure, root closure, and complete integral closure of a Puiseux algebra, and use such description to provide a class of antimatter Puiseux algebras (i.e., Puiseux algebras containing no irreducibles).