On semigroup algebras with rational exponents

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 is called a Puiseux algebra and denoted by 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 and 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 we use this description to provide a class of antimatter Puiseux algebras (i.e., Puiseux algebras containing no irreducibles).