Phase hyperfield - Knowlpedia

The phase hyperfield is the quotient hyperfield

\mathbb P=\mathbb C/\mathbb R_{>0}.

Its elements are $0$ together with the complex phases $S^1$ . Multiplication is ordinary multiplication of phases, while the hyper-sum of two phases is the set of phases of all sums of positive real multiples of representatives.

For distinct non-antipodal $u,v\in S^1$ , the sum $u\boxplus v$ is the open shorter arc from $u$ to $v$ . If $v=-u$ , then

u\boxplus(-u)=\{0,u,-u\}.

The quotient map $\mathbb C\to\mathbb P$ records the phase of a nonzero complex number and is a weak hyperfield homomorphism.

References

Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034.
Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204.