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\begin{document}

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\section*{hyperring}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{hyperfield_extension_of_field_with_one_element}{Hyperfield extension of field with one element}\dotfill \pageref*{hyperfield_extension_of_field_with_one_element} \linebreak
\noindent\hyperlink{the_signature_hyperfield_}{The signature hyperfield $\mathbf{S}$}\dotfill \pageref*{the_signature_hyperfield_} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{hyperring} is like a [[ring]] not with an underlying [[abelian group]] but an underlying [[canonical hypergroup]].

It is a [[hypermonoid]] with additional ring-like [[stuff, structure, property|structure and properties]].

This means that in a hyperring $R$ addition is a multi-valued operation.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{hyperring} is a non-empty [[set]] $R$ equipped with a hyper-addition $+ : R\times R \to P^*(R)$ (where $P^*(R)$ is the set of non-empty subsets) and a multiplication $\cdot : R \times R \to R$ and with elements $0,1 \in R$ such that

\begin{enumerate}%
\item $(R,+)$ is a [[canonical hypergroup]];


\item $(R,\cdot)$ is a [[monoid]] with identity element $1$;


\item $\forall r,s \in R : r(s+t) = r s + r t$ and $(s + t) r = s r + t r$;


\item $\forall r \in R : r \cdot 0 = 0 \cdot r = 0$;


\item $0 \neq 1$.



\end{enumerate}
We can form many examples of hyperrings by quotienting a ring $R$ by a subgroup $G \subset R^{\times}$ of its multiplicative group.

A \textbf{morphism of hyperrings} is a map $f : R_1 \to R_2$ such that

\begin{enumerate}%
\item $\forall a,b \in R_1 : f(a + b) \subset f(a) + f(b)$;


\item $\forall a,b\in R_1 : f(a \cdot b) = f(a) \cdot f(b)$.



\end{enumerate}
A \textbf{hyperfield} is a hyperring for which $(R - \{0\}, \cdot)$ is a [[group]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{hyperfield_extension_of_field_with_one_element}{}\subsubsection*{{Hyperfield extension of field with one element}}\label{hyperfield_extension_of_field_with_one_element}

The \textbf{hyperfield extension of the [[field with one element]]} is

\begin{displaymath}
\mathbf{K} := (\{0,1\}, +, \cdot)
\end{displaymath}
with additive neutral element $0$ and the hyper-addition rule

\begin{displaymath}
1 + 1 = \{0,1\}
  \,.
\end{displaymath}
This is to be thought of as the hyperring of [[integer]]s modulo the relation ``is 0 or not 0'': think of $0 \in \mathbf{K}$ as being the integer 0 and of $1 \in \mathbf{K}$ as being \emph{any} non-zero integer, then the addition rule says that 0 plus any non-zero integer is non-zero, and that the sum of a non-zero integer with another non-zero integer is either zero or non-zero.

\hypertarget{the_signature_hyperfield_}{}\subsubsection*{{The signature hyperfield $\mathbf{S}$}}\label{the_signature_hyperfield_}

Let $\mathbf{S} = \{0,1,-1\}$ be the hyperfield with multiplication induced from $\mathbb{Z}$ and with addition given by 0 being the additive unit and the laws

\begin{itemize}%
\item $1+1 = \{1\}$;


\item $-1 + -1 = \{-1\}$


\item $1 + -1 = \{-1, 0, 1\}$.



\end{itemize}
This we may think of as being the hyperring of [[integer]]s modulo the relation ``is positive or negative or 0'': think of $1$ as being any positive integer, $0$ as being the integer $0$ and $-1$ as being any negative integer. Then the hyper-addition law above encodes how the signature of integers behaves under addition.

\textbf{Proposition}

To each element, $\phi$, of $Hom(\mathbb{Z}[X], \mathbf{S})$ there corresponds an extended real number, $Re(\phi) \in [-\infty, \infty]$ given as a Dedekind cut. This is a surjective mapping. The inverse image of each real algebraic number contains three elements, while that of a nonalgebraic number is a singleton. For real algebraic $\alpha$, the three homomorphisms from $\mathbb{Z}[X]$ to $\mathbf{S}$ are

\begin{displaymath}
P(T) \mapsto \underset{\epsilon \to 0+} {lim} sign P(\alpha + t \epsilon), t \in \{-1, 0, 1\}.
\end{displaymath}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[hypergroup]]


\item [[canonical hypergroup]]


\item [[hypermonoid]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion of hyperring and hyperfield is due to Marc Krasner:

\begin{itemize}%
\item M. Krasner, \emph{Approximation des corps valu\'e{}s complets de caract\'e{}ristique $p\neq 0$ par ceux de caract\'e{}ristique 0} , pp.126-201 in \emph{Colloque d' Alg\'e{}bre Sup\'e{}rieure (Bruxelles), 1956} , Ceuterick Louvain 1957.

\end{itemize}
Another early reference is

\begin{itemize}%
\item D. Stratigopoulos, \emph{Hyperanneaux non commutatifs: Hyperanneaux, hypercorps, hypermodules, hyperespaces vectoriels et leurs propri\'e{}t\'e{}s \'e{}l\'e{}mentaires} (French) C. R. Acad. Sci. Paris S\'e{}r. A-B 269 (1969) A489--A492.

\end{itemize}
Modern applications in connection to the [[field with one element]] are discussed in

\begin{itemize}%
\item [[Alain Connes]], [[Caterina Consani]], \emph{The hyperring of ad\`e{}le classes} (\href{http://arxiv.org/abs/1001.4260}{arXiv:1001.4260})


\item [[Alain Connes]], [[Caterina Consani]], \emph{From monoids to hyperstructures: in search of an absolute arithmetic} (\href{http://arxiv.org/abs/1006.4810}{arXiv:1006.4810})



\end{itemize}
An overview is in

\begin{itemize}%
\item Jaiung Jun, \emph{Algebraic Geometry over Hyperrings} , arXiv:1512.04837 (2015). (\href{http://arxiv.org/abs/1512.04837}{abstract})

\end{itemize}
[[!redirects hyperring]] [[!redirects hyperrings]] [[!redirects hyper-ring]] [[!redirects hyper-rings]]

[[!redirects hyperfield]] [[!redirects hyperfields]] [[!redirects hyper-field]] [[!redirects hyper-fields]]



\end{document}