Hyperstructures and Idempotent Semistructures

[摘要] 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 $epsilon:mathfrak{m}_K/mathfrak{m}_K^{n+1}ightarrow mathcal{O}_K/mathfrak{m}_K^n$ 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 ${0}cup{u^kmid kinmathbb{Z}}$ 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 $xin H$. We say a hyperfield $K$ is selective if $1+1-1-1=1-1$ and for all $x,yin K$ one has either $xin x+y$ 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 $phiinmathrm{Ext}^1(Gamma,H^imes)$.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.

[发布机构] Johns Hopkins University

[效力级别] multirings

[关键词] semirings;multirings;Mathematics