Hyperfields

definitions

Hyperfields are defined similarly to fields, but addition is instead a hyperoperation, or a multi-valued function.

To formalize this, we can introduce the notion of a hypergroup: a hypergroup is a set $G$ along with an associative binary hyperoperation $\boxplus : G \times G \to 2^G \setminus \{\varnothing\}$ such that:

1. (identity) there is an element $0 \in G$ such that $0 \boxplus x = \{x\}$ for all $x \in G$,
2. (inverse) for all $a \in G$, there is an element $-a \in G$ such that $0 \in a \boxplus -a$,
3. (reversibility) for all $a, b, c \in G$, $a \in b \boxplus c$ if and only if $c \in a \boxplus -b$.

Now, we can define a hyperfield as the data $(F, 0, 1, \boxplus, \cdot)$ where

1. $(F, 0, \boxplus)$ is a commutative hypergroup,
2. $(F \setminus \{0\}, 1, \cdot)$ is an abelian group,
3. for all $a \in F$, $a \cdot 0 = 0$,
4. for all $a, b, c \in F$, $a \cdot (b \boxplus c) = (a \cdot b) \boxplus (a \cdot c)$,
   • here, $a \cdot (b \boxplus c) = \{a \cdot d \mid d \in b \boxplus c\}$
5. $0 \neq 1$.

Finally, a (weak) homomorphism between two hyperfields is a map $\phi: F_1 \to F_2$ such that

1. $\phi(0) = 0$ and $\phi(1) = 1$,
2. for all $a,b \in F_1$, $\phi(a \cdot b) = \phi(a) \cdot \phi(b)$,
3. for all $a,b \in F_1$, $\phi(a \boxplus b) \subseteq \phi(a) \boxplus \phi(b)$
   • Note the use of $\subseteq$ rather than $=$ here! While a weak homomorphism only requires inclusion, a strong homomorphism requires equality.

Naturally, a homomorphism is an isomorphism if its inverse is a homomorphism as well.

From these definitions, we can begin to discuss the number of hyperfields of a given order up to isomorphism. Complete classification ends up not being a trivial task, but it is doable for small orders.

results

As part of my research, I developed a computer program to generate and encode hyperfields of any given order up to isomorphism class. Doing so, I was able to produce a novel result of the number of hyperfields of order 7 up to isomorphism class. Here are the results for the first few orders:

To calculate this number, I categorized hyperfields $F$ by their multiplicative group (a similar concept is that of the underlying monoid; see [3]). Doing so allows for the use of the fundamental theorem of abelian groups, simplifying classification since an isomorphism of hyperfields must induce an isomorphism of multiplicative groups. It may be more intuitive now why the number of hyperfields of order 6 is less than the number of hyperfields of order 5: an order 6 hyperfield necessitates a multiplicative group of $\mathbb Z_5$, whereas an order 5 hyperfield could have multiplicative group $\mathbb Z_4$ or $\mathbb Z_2 \times \mathbb Z_2$.

future directions

The program I wrote is not very optimized; computing the result for order 7 is already quite slow and order 8 is out of the question. I believe this is due to two things—the program is written in Java, and the isomorphism calculations are not particularly efficient. In the future, I hope to optimize the program, most likely in a different language (perhaps Haskell / Rust, or formalizing with Lean).

Additionally, an open question asks about the proportion of hyperfields that can be expressed a quotient of a field by some of its multiplicative subgroups, and what this proportion limits to as the order tends to infinity. If possible, I would like to find a way to encode this information in the program as well.

references