# Contents

## Idea

A hyperring is like a ring not with an underlying abelian group but an underlying canonical hypergroup.

It is a hypermonoid with additional ring-like structure and properties.

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

## Definition

A 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

1. $(R,+)$ is a canonical hypergroup;

2. $(R,\cdot)$ is a monoid with identity element $1$;

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

4. $\forall r \in R : r \cdot 0 = 0 \cdot r = 0$;

5. $0 \neq 1$.

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

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

1. $\forall a,b \in R_1 : f(a + b) \subset f(a) + f(b)$;

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

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

## Examples

### Hyperfield extension of field with one element

The hyperfield extension of the field with one element is

$\mathbf{K} := (\{0,1\}, +, \cdot)$

with additive neutral element $0$ and the hyper-addition rule

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

This is to be thought of as the hyperring of integers 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 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.

### The signature hyperfield $\mathbf{S}$

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

• $1+1 = \{1\}$;

• $-1 + -1 = \{-1\}$

• $1 + -1 = \{-1, 0, 1\}$.

This we may think of as being the hyperring of integers 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.

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

$P(T) \mapsto \underset{\epsilon \to 0+} {lim} sign P(\alpha + t \epsilon), t \in \{-1, 0, 1\}.$

## References

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

• M. Krasner, Approximation des corps valués complets de caractéristique $p\neq 0$ par ceux de caractéristique 0 , pp.126-201 in Colloque d’ Algébre Supérieure (Bruxelles), 1956 , Ceuterick Louvain 1957.

Another early reference is

• D. Stratigopoulos, Hyperanneaux non commutatifs: Hyperanneaux, hypercorps, hypermodules, hyperespaces vectoriels et leurs propriétés élémentaires (French) C. R. Acad. Sci. Paris Sér. A-B 269 (1969) A489–A492.

Modern applications in connection to the field with one element are discussed in

An overview is in

• Jaiung Jun, Algebraic Geometry over Hyperrings , arXiv:1512.04837 (2015). (abstract)

Revised on March 30, 2016 08:32:39 by Thomas Holder (176.0.29.143)