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 M₁ and M₂ 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).

摘要

在本文中，一个半群代数由多项式表达式组成，该多项式表达式具有域F中的系数和加性子类M中的指数${\mathbb{问}}_{\ge 0}$被称为 Puiseux 代数并表示为$F[米].$在这里，我们研究了 Puiseux 代数的原子结构。首先，我们回答了 Puiseux 代数类的同构问题，也就是说，我们证明对于域F，如果两个 Puiseux 代数$F[{米}_1]$ 和 $F[{米}_2]$是同构的，则幺半群M ₁和M ₂必须是同构的。然后我们构造了满足以下众所周知的原子性质的三类 Puiseux 代数：ACCP 性质、有界分解性质和有限分解性质。我们证明了有界分解 Puiseux 代数具有长度集的极值系统，这使我们能够证明 Puiseux 代数不能通过它们的长度算术来确定（直到同构）。最后，我们给出了一个Puiseux代数的半正规闭包、根闭包和完全积分闭包的完整描述，并且我们使用这个描述来提供一类反物质Puiseux代数（即，不包含不可约数的Puiseux代数）。