nLab hyperring

Idea

A hyperring is like a ring but not with an underlying abelian group but with an underlying canonical hypergroup, hence it is a hypermonoid with additional ring-like structure. This means that addition in a hyperring $R$ is a multi-valued function. A hyperfield is a hyperring where every nonzero element has a multiplicative inverse.

From Viro 2010:

“Krasner, Marshall, Connes and Consani and the author came to hyperfields for different reasons, motivated by different mathematical problems, but we came to the same conclusion: the hyperrings and hyperfields are great, very useful and very underdeveloped in the mathematical literature… Probably, the main obstacle for hyperfields to become a mainstream notion is that a multivalued operation does not fit to the tradition of set-theoretic terminology, which forces to avoid multivalued maps at any cost. I believe the taboo on multivalued maps has no real ground, and eventually will be removed. Hyperfields, as well as multigroups, hyperrings and multirings, are legitimate algebraic objects related in many ways to the classical core of mathematics… I believe hyperfields are to displace the tropical semifield in the tropical geometry. They suit the role better. In particular, with hyperfields the varieties are defined by equations, as in other branches of algebraic geometry.”

Definition

A hyperring is a non-empty set $R$ equipped with

1. a hyper-addition $+ \,\colon\, R\times R \longrightarrow P^*(R)$ (where $P^*(R)$ is the set of non-empty subsets)

2. a multiplication $\cdot \,\colon\, R \times R \longrightarrow R$

3. chosen elements $0,1 \,\in\, R$

such that

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

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

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

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

(e.g. Nakassis 1987 Def. 4, Davvaz & Leoreanu-Fotea 2007 Def. 3.1.1; other authors require moreover that $0 \neq 1$, e.g. Connes & Consani 2011 Def. 2.1, Jun 2015 Def. 2.2)

A homomorphism of hyperrings is a map $f \,\colon\, R_1 \to R_2$ such that

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

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

A hyperfield is a hyperring for which $(R \setminus \{0\}, \cdot)$ is a group and $0 \ne 1$.

Examples

Equivalence classes in a ring

We can form many examples of hyperrings by quotienting a ring $R$ by some subgroup $G \subseteq R^{\times}$ of its group of units such that $r G = G r$ for all $r \in R$. In more detail, an element of the hyperring $R/G$ is given by an equivalence class of elements of $R$, where $x \sim y$ if and only if $x = y g$ for some $g \in G$. Thanks to the condition $r G = G r$, multiplication descends straightforwardly from $R$ to $R/G$, while addition become multivalued: the sum of equivalence classes $[x]$ and $[y]$ is the set of all equivalence classes $[x + y]$.

In particular, if $R$ is a hyperfield so is $R/G$.

We say a hyperring is doubly distributive if obeys the familiar identity

$$(a + b)(c + d) = a c + a d + b c + b d$$

Not every hyperring is doubly distributive (see the example of the triangle hyperring below): in general, we merely have

$$(a + b)(c + d) \subseteq a c + a d + b c + b d$$

However, every hyperring $R/G$ obtained as a quotient of a ring $R$ by the above construction is doubly distributive.

The Krasner hyperfield $\mathbb{K}$

The Krasner hyperfield is the set $\mathbb{K} = \{0,1\}$ with multiplication given by

• $0 \cdot x = 0$

• $1 \cdot x = x$

and with addition given by

• $0 + x = x$

• $1+1 = \{0,1\}$

$\mathbb{K}$ is the quotient of the field of rational numbers or real numbers by its multiplicative subgroup consisting of all nonzero numbers: $0$ is the equivalence class containing $0$, while $1$ is the equivalence class containing all nonzero numbers. The hyper-addition law above encodes how such equivalence classes behave under addition: for example, the sum of two nonzero numbers can be zero or nonzero.

The sign hyperfield $\mathbb{S}$

The sign hyperfield is the set $\mathbb{S} = \{0,1,-1\}$ with multiplication given by

• $0 \cdot x = 0$

• $1 \cdot x = 1$

• $-1 \cdot -1 = 1$

and with addition given by

• $0 + x = x$

• $1+1 = \{1\}$

• $-1 + -1 = \{-1\}$

• $1 + -1 = \{-1, 0, 1\}$.

$\mathbb{S}$ is the quotient of the field of rational numbers or real numbers by its multiplicative subgroup of positive numbers: $1$ is the equivalence class containing the positive numbers, $0$ is the equivalence class containing $0$, and $-1$ as the equivalence class of the negative numbers. Again, the hyper-addition law encodes how such equivalence classes behave under addition: most notably, the sum of a positive and a negative number can be positive, zero, or negative.

Proposition

To each element, $\phi$, of $Hom(\mathbb{Z}[X], \mathbb{S})$ 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 $\mathbb{S}$ are

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

The tropical hyperfield $\mathbb{T}$

The tropical hyperfield has as its underlying set $\mathbb{T} = [0,\infty]$. The product of $a, b \in \mathbb{T}$ is defined to be the usual sum of nonnegative numbers for $a, b\in [0,\infty)$, and we define $a \cdot \infty = \infty$. The sum of $a,b \in \mathbb{T}$ is defined to be $a \max b$ when $a \ne b$, but when $a = b$ it is defined to be the set $[0,a]$.

Proposition. For any commutative ring $R$, hyperring homomorphisms $\phi \colon R \to \mathbb{T}$ are the same as nonarchimedean valuations $\phi \colon R \to [0,\infty]$.

This was observed first by Viro (2010), but a proof can be found in Lorscheid’s Tropical geometry over the tropical hyperfield, Theorem 2.2.

The triangle hyperfield $\Delta$

The triangle hyperfield $\Delta$ is the set $\mathbb{R}^+ = [0,\infty)$ with its usual multiplication, where the sum of $a$ and $b$ is the set of all $c \in \mathbb{R}^+$ such $a,b,c$ are the lengths of the sides of a triangle in the plane (allowing sides of length zero). Viro showed $\Delta$ is a hyperfield in Theorem 5.4 of his paper on hyperfields in tropical geometry. This hyperfield is not doubly distributive, because

$$(2 + 1) \cdot (2 + 1) = [1,3] \cdot [1,3] = [1,9]$$

while

$$2 \cdot 2 + 2 \cdot 1 + 1 \cdot 2 + 1 \cdot 1 = 4 + 2 + 2 + 1 = [0,9]$$

Related entries

• ring

• hypermagma

• hypermonoid

• hypergroup

• hypergraph

• canonical hypergroup

References

The notion of hyperring and hyperfield is due to:

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

Another early reference is

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

See also:

• Marc Krasner: A class of hyperrings and hyperfields, International Journal of Mathematics and Mathematical Sciences (1983) [doi:10.1155/S0161171283000265]

Review:

• Anastase Nakassis: Recent results in hyperring and hyperfield theory, International Journal of Mathematics and Mathematical Sciences (1987) [doi:10.1155/S0161171288000250]

Monographs:

• Bijan Davvaz, Violeta Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press (2007) [pdf]

• Bijan Davvaz, Violeta Leoreanu-Fotea, Krasner Hyperring Theory, World Scientific (2024) [doi:10.1142/13652]

Applications to the geometry over the “field with one element”:

• Alain Connes, Caterina Consani, The hyperring of adèle classes, Journal of Number Theory 131 2 (2011) 159-194 [arXiv:1001.4260, doi:10.1016/j.jnt.2010.09.001]

• Alain Connes, Caterina Consani: From monoids to hyperstructures: in search of an absolute arithmetic [arXiv:1006.4810]

On algebraic geometry over hyperrings:

• Jaiung Jun, Algebraic geometry over hyperrings, Advances in Mathematics 323 (2018) 142-192 [arXiv:1512.04837, doi:10.1016/j.aim.2017.10.043]

On the tropical hyperfield:

• Oleg Viro: Hyperfields for tropical geometry I. hyperfields and dequantization [arXiv:1006.3034]

• Oliver Lorscheid, Tropical geometry over the tropical hyperfield [arXiv:1907.01037]

category: algebra