Knowlpedia

Definition

Doubly distributive hyperfield

A hyperfield in which the product of two binary hypersums equals the four-term hyper-sum of pairwise products.

Definition

A FF is doubly distributive if

(a⊞b)(c⊞d)=ac⊞ad⊞bc⊞bd(a\boxplus b)(c\boxplus d) =ac\boxplus ad\boxplus bc\boxplus bd

as subsets of FF for every a,b,c,d∈Fa,b,c,d\in F. On the left, multiplication of subsets means

XY={xy:x∈X, y∈Y}.XY=\{xy:x\in X,\ y\in Y\}.

Ordinary distributivity only distributes multiplication by one element over one hypersum. Double distributivity is the stronger assertion that distributing the product of two hypersums introduces no extra or missing values.

Examples

, the , the , and the are doubly distributive. Every doubly distributive hyperfield is , but the converse fails.

References

Nathan Bowler and Ting Su, Classification of doubly distributive skew hyperfields and stringent hypergroups, Definition 2.8.