Hyperstructures and Idempotent Semistructures
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Date
2015-03-15
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Johns Hopkins University
Abstract
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.
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Keywords
semirings, multirings