关于超域上多项式超结构中超乘法结合性的几个结果
A few results on associativity of hypermultiplications in polynomial hyperstructures over hyperfields
摘要：
在Baker和Lorscheid的论文中，他们引入了一种新的超结构：超域$\mathbb{F}$上的多项式超结构Poly $(\mathbb{F})$。在这篇文章中，作者着重研究了超乘法在这些超结构中的结合性，并给出了一些基本命题。作者还展示了具有非结合超乘法的超域上的多项式超结构的例子。然后证明了Poly $(\mathbb{T})$中的超乘法对线性多项式是结合的，但一般情况下不是结合的。此外，他还证明了如果$1\boxplus_{\mathbb{F}}1$不是超域$\mathbb{F}:=(\mathbb{F},\odot,\boxplus_{\mathbb{F}},1,0)$的单例，Poly $(\mathbb{F})$中的超乘法就不是结合的。
英文摘要：
In Baker and Lorscheid's paper, they introduce a new hyperstructure: the polynomial hyperstructure Poly$(\mathbb{F})$ over a hyperfield $\mathbb{F}$. In this work, the author focuses on associativity of hypermultiplications in those hyperstructures and gives elementary propositions. The author also shows examples of polynomial hyperstructures over hyperfields with non-associative hypermultiplications. Then, he proves that though the hypermultiplication in Poly$(\mathbb{T})$ is associative for linear polynomials, it is not associative in general. Moreover, he shows that if $1\boxplus_{\mathbb{F}}1$ is not a singleton for hyperfield $\mathbb{F}:=(\mathbb{F},\odot,\boxplus_{\mathbb{F}},1,0)$, the hypermultiplication in Poly$(\mathbb{F})$ is not associative.
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