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.
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
$(R,+)$ is a canonical hypergroup;
$(R,\cdot)$ is a monoid with identity element $1$;
$\forall r,s \in R : r(s+t) = r s + r t and (s + t) r = s r + t r$;
$\forall r \in R : r \cdot 0 = 0 \cdot r = 0$;
$0 \neq 1$.
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
$\forall a,b \in R_1 : f(a + b) \subset f(a) + f(b)$;
$\forall a,b\in R_1 : f(a \cdot b) = f(a) \cdot f(b)$.
A hyperfield is a hyperring for which $(R - \{0\}, \cdot)$ is a group.
The hyperfield extension of the field with one element is
with additive neutral element $0$ 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 $0 \in \mathbf{K}$ as being the integer 0 and of $1 \in \mathbf{K}$ 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.
Let $\mathbf{S} = \{0,1,-1\}$ be the hyperfield with multiplication induced from $\mathbb{Z}$ and with addition given by 0 being the additive unit and the laws
$1+1 = \{1\}$;
$-1 + -1 = \{-1\}$
$1 + -1 = \{-1, 0, 1\}$.
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, $\phi$, of $Hom(\mathbb{Z}[X], \mathbf{S})$ 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 $\mathbf{S}$ 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
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)