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
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.
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.
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 maybe (?) due to Krasner. An early reference is
Modern applications in connection to the field with one element are discussed in