Krasner hyperfield - Knowlpedia

Example

The Krasner hyperfield is the two-element set $\mathbb K=\{0,1\}$ with ordinary multiplication, $0$ as additive identity, and

1\boxplus1=\{0,1\}.

Together with $0\boxplus x=\{x\}$ , this determines all of its hyperaddition and makes $\mathbb K$ a hyperfield.

Quotient realization

If $F$ is a field with at least three elements, then

F/F^\times\cong\mathbb K.

There are two multiplicative orbits, $0$ and the nonzero elements. A sum of two nonzero representatives can be zero, and suitable representatives also give a nonzero sum, producing both elements of $1\boxplus1$ .

For $F=\mathbb F_2$ , however, the quotient sum of the sole nonzero orbit with itself is only $\{0\}$ ; the “at least three elements” hypothesis is therefore essential for this realization.

Boolean comparison

The Boolean semifield also has carrier $\{0,1\}$ and ordinary multiplication, but $1+1=1$ there. The Krasner hyperfield has a genuinely multivalued sum containing both $0$ and $1$ , so the two structures are not isomorphic and should not be conflated.

References

Alain Connes and Caterina Consani, “The hyperring of adèle classes,” Journal of Number Theory 131 (2011), 159–194. arXiv:1001.4260. Relevant: the Krasner hyperfield $\mathbb K$ and quotient hyperrings.
Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: Example 2.4.