nLab
hyperring

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 RR addition is a multi-valued operation.

Definition

A hyperring is a non-empty set RR equipped with a hyper-addition +:R×RP *(R)+ : R\times R \to P^*(R) (where P *(R)P^*(R) is the set of non-empty subsets) and a multiplication :R×RR\cdot : R \times R \to R and with elements 0,1R0,1 \in R such that

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

  2. (R,)(R,\cdot) is a monoid with identity element 11;

  3. r,sR:r(s+t)=rs+rtand(s+t)r=sr+tr\forall r,s \in R : r(s+t) = r s + r t and (s + t) r = s r + t r;

  4. rR:r0=0r=0\forall r \in R : r \cdot 0 = 0 \cdot r = 0;

  5. 010 \neq 1.

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

A morphism of hyperrings is a map f:R 1R 2f : R_1 \to R_2 such that

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

  2. a,bR 1:f(ab)=f(a)f(b)\forall a,b\in R_1 : f(a \cdot b) = f(a) \cdot f(b).

A hyperfield is a hyperring for which (R{0},)(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},+,) \mathbf{K} := (\{0,1\}, +, \cdot)

with additive neutral element 00 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 0K0 \in \mathbf{K} as being the integer 0 and of 1K1 \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 S\mathbf{S}

Let S={0,1,1}\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}-1 + -1 = \{-1\}

  • 1+1={1,0,1}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 11 as being any positive integer, 00 as being the integer 00 and 1-1 as being any negative integer. Then the hyper-addition law above encodes how the signature of integers behaves under addition.

Proposition

To each element, ϕ\phi, of Hom([X],S)Hom(\mathbb{Z}[X], \mathbf{S}) there corresponds an extended real number, Re(ϕ)[,]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 α\alpha, the three homomorphisms from [X]\mathbb{Z}[X] to S\mathbf{S} are

P(T)limϵ0+signP(α+tϵ),t{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

Revised on July 19, 2010 22:09:02 by Toby Bartels (64.89.48.241)