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 addition is a multi-valued operation.
A hyperring is a non-empty set equipped with a hyper-addition (where is the set of non-empty subsets) and a multiplication and with elements such that
is a canonical hypergroup;
is a monoid with identity element ;
We can form many examples of hyperrings by quotienting a ring by a subgroup of its multiplicative group.
A morphism of hyperrings is a map such that
A hyperfield is a hyperring for which is a group.
Hyperfield extension of field with one element
The hyperfield extension of the field with one element is
with additive neutral element and the hyper-addition rule
This is to be thought of as the hyperring of integers modulo the relation “is 0 or not 0”: think of as being the integer 0 and of 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
Let be the hyperfield with multiplication induced from and with addition given by 0 being the additive unit and the laws
This we may think of as being the hyperring of integers modulo the relation “is positive or negative or 0”: think of as being any positive integer, as being the integer and as being any negative integer. Then the hyper-addition law above encodes how the signature of integers behaves under addition.
To each element, , of there corresponds an extended real number, 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 , the three homomorphisms from to are
The notion of hyperring and hyperfield is due to Marc Krasner:
- M. Krasner, Approximation des corps valués complets de caractéristique 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)