Hyperstructures and Idempotent Semistructures

Embargo until
Journal Title
Journal ISSN
Volume Title
Johns Hopkins University
Much of this thesis concerns hypergroups, multirings, and hyperfields. These are analogous to abelian groups, rings, and fields, but have a multivalued addition operation. M. Krasner introduced the notion of a valued hyperfield; The prototypical example is $K/(1+\mathfrak{m}_K^n)$ where $K$ is a local field. P. Deligne introduced a category of triples whose objects have the form $\mathrm{Tr}_n(K)=(\mathcal{O}_K/\mathfrak{m}_K^n,\mathfrak{m}_K/\mathfrak{m}_K^{n+1},\epsilon)$ where $\epsilon:\mathfrak{m}_K/\mathfrak{m}_K^{n+1}\rightarrow \mathcal{O}_K/\mathfrak{m}_K^n$ is the obvious map. In this thesis I relate the category of discretely valued hyperfields to Deligne's category of triples. An extension of a local field is arithmetically profinite if the upper ramification subgroups are open. Given such an extension $L/K$, J.P. Wintenberger defined the norm field $X_K(L)$ as the inverse limit of the finite subextensions of $L/K$ along the norm maps. Wintenberger has defined an addition operation on $X_K(L)$, and shown that $X_K(L)$ is a local field of finite characteristic. Using Deligne's triples, I have given a new proof of Wintenberger's characterization of its Galois group. The semifield $\mathbb{Z}_{\max}$ is defined as $\{0\}\cup\{u^k\mid k\in\mathbb{Z}\}$ with addition given by $u^m+u^n=u^{\max(m,n)}$. An extension of $\mathbb{Z}_{\max}$ is a semifield containing $\mathbb{Z}_{\max}$. The extension is finite if $S$ is finitely generated as a $\mathbb{Z}_{\max}$-semimodule. In this thesis I classify the finite extensions of $\mathbb{Z}_\mathrm{max}$. There are two previously known methods for constructing a hypergroup from a totally ordered set. In this thesis I generalize these to a family of constructions parametrized by hypergroups $H$ satisfying $x-x=H$ for all $x\in H$. We say a hyperfield $K$ is selective if $1+1-1-1=1-1$ and for all $x,y\in K$ one has either $x\in x+y$ or $y=x+y$. In this thesis, I show that a selective hyperfield is characterized by a totally ordered group $\Gamma$, a hyperfield $H$ satisfying $1-1=H$, and an extension $\phi\in\mathrm{Ext}^1(\Gamma,H^\times)$. We say a triple of elements $(x,y,z)$ of an idempotent semiring is a corner triple if $x+y=y+z=x+z$. We say an idempotent semiring is regular if whenever $(x,y,a)$ and $(z,w,a)$ are corner triples, there exists $b$ such that $(x,z,b)$ and $(y,w,b)$ are also corner triples. I prove in this thesis that the category of regular idempotent semirings is a reflective subcategory of the category of multirings.
semirings, multirings