Knowlpedia

Example

Sign hyperfield

The three-element hyperfield recording only whether a real number is negative, zero, or positive.

The sign hyperfield is S={0,1,−1}\mathbb S=\{0,1,-1\} with the usual multiplication of signs and hyperaddition determined by

1⊞1={1},(−1)⊞(−1)={−1},1⊞(−1)={−1,0,1}.1\boxplus1=\{1\},\qquad (-1)\boxplus(-1)=\{-1\},\qquad 1\boxplus(-1)=\{-1,0,1\}.

Also 0⊞x={x}0\boxplus x=\{x\}. It is a hyperfield whose elements retain only sign information.

Quotient realization

The sign map realizes

S≅R/R>0.\mathbb S\cong\mathbb R/\mathbb R_{>0}.

Positive rescaling partitions R\mathbb R into the negative, zero, and positive orbits. Two numbers of the same nonzero sign have a sum of that sign, whereas a positive and a negative number can sum to a number of any sign. This exactly produces the displayed hyperaddition.

Morphism from the real field

The map

sgn⁡:R⟶S\operatorname{sgn}:\mathbb R\longrightarrow\mathbb S

is a weak hyperfield homomorphism: it preserves zero, one, and multiplication, and sgn⁡(x+y)∈sgn⁡(x)⊞sgn⁡(y)\operatorname{sgn}(x+y)\in \operatorname{sgn}(x)\boxplus\operatorname{sgn}(y). It is not strong, since a fixed pair x,yx,y has one sign for its sum while the target hyper-sum can contain three signs.

References
  1. Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: the hyperfield of signs and factor hyperfields.
  2. Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: Example 2.6 and oriented matroids over the sign hyperfield.