A few results on associativity of hypermultiplications in polynomial hyperstructures over hyperfields

Abstract

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.