nLab
hyperring

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. $\forall r,s\in R:r(s+t)=rs+rt\mathrm{and}(s+t)r=sr+tr$;

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

5. $0\ne 1$.

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

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

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

$$1+1=\{0,1\}.$$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 $0\in K$ as being the integer 0 and of $1\in K$ 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 $S$

Let $S=\{0,1,-1\}$ 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, $\varphi$, of $\mathrm{Hom}(\mathbb{Z}[X],S)$ there corresponds an extended real number, $\mathrm{Re}(\varphi )\in [-\mathrm{\infty },\mathrm{\infty }]$ 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 $S$ are

$$P(T)\mapsto \underset{ϵ\to 0+}{\lim }\mathrm{sign}P(\alpha +tϵ),t\in \{-1,0,1\}.$$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 maybe (?) due to Krasner. An 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

• Alain Connes, Caterina Consani?, The hyperring of adèle classes (arXiv:1001.4260)

• Alain Connes, Caterina Consani?, From monoids to hyperstructures: in search of an absolute arithmetic (arXiv:1006.4810)