Question

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a weakening of the ring concept, but where the addition is allowed to be multivalued. Indeed the additive part of a hyperring forms a 'canonical hypergroup'.

A canonical hypergroup is a set, $H$, equipped with a commutative binary operation,

$$ + : H \times H \to P^*(H) $$

taking values in non-empty subsets of $H$, and a zero element $0 \in H$, such that

1. $+$ is associative (extended to allow addition of subsets of $H$);
2. $0 + x = {{x}} = x + 0, \forall x \in H$;
3. $\forall x \in H, \exists ! y \in H$ such that $0 \in x + y$ (we denote this $y$ as $-x$);
4. $\forall x, y, z \in H, x \in y + z$ implies $z \in x - y$ (where $x - y$ means $x + (-y)$ as usual).

(NB: $x$ may be written for the singleton {$x$}.)

I know that hyperrings occur whenever a ring is quotiented by a subgroup of its multiplicative group, but I'd like to know more about where and how hyperrings and hypergroups have cropped up in different branches of mathematics. How is a canonical hypergroup to be thought of as canonical? Are noncanonical hypergroups important? Is there a category theoretic way to see these hyperstructures as natural?

Answer 1

While I don't know much about hyperstructures other than hypergroups, I know it is hard to study the history behind them because of the non-consistent terminology attributed to these objects by different authors in different periods. I will say something about hypergroups and hopefully some specialist can come and give better insight.

First some physical intuition for finite commutative hypergroups that I found useful: the simplest way to think of them is to think of a collection of particle types $\{c_0,c_1,\cdots,c_n\}$ where two particles can collide to form a third, however not in a definite manner. Let the structure constants $n_{ij}^k$ denote the probability that $c_i+c_j\rightarrow c_k$. Now assume $c_0$ denotes photons and that they get absorbed in every collision. Also assume that for each particle there is a unique antiparticle so that their collision is likely to produce a photon with non-zero probability.

So coming to the actual definitions, call a generalized hypergroup a pair $(\mathcal K, \mathcal A)$ where $\mathcal A$ is a *-algebra with unit $c_0$ over $\mathbb C$ and $\mathcal K =\{c_0,c_1\dots,c_n\}$ is a basis of $\mathcal A$ with $\mathcal K ^*=\mathcal K$ for which the structure constants $n_{ij}^k$ defined by $$c_ic_j=\sum_k n_{ij}^k c_k$$ satisfy the conditions $c_i^*=c_j \iff n_{ij}^0>0$ and $c_i^*\neq c_j \iff n_{ij}^0 =0$.

$(\mathcal K,\mathcal A)$ is called Hermitian if $c_i^*=c_i$ for all $i$, commutative if $c_ic_j=c_jc_i$ for all $i,j$, real if $n_{ij}^k\in \mathbb R$ for all $i,j,k$, positive if $n_{ij}^k\geq 0$ for all $i,j,k$ and normalized if $\sum_k n_{ij}^k =1$ for all $i,j$. A hypergroup is a generalized hypergroup which is both positive and normalized (if positive is replaced by real you get what's called a signed hypergroup).

Now coming to canonical hypergroups, it is easy to see that associated to any hypergroup you have a new one where the hyperoperation is defined by $$c_i\circ c_j=\{c_k \quad | \quad n_{ij}^k\neq 0\}$$ and it is in this sense that they are to be thought of as canonical, and if you accept canonical hypergroups as important then the non-canonical ones are too.

All of the above is written from Wildberger's "Finite commutative hypergroups and applications from group theory to conformal field theory", and let me add here for the ones who can not reach the article a list of mentioned mathematical objects/theories that are very close to the concept of a hypergroup and have been studied under a plethora of different names: Kawada's work on C-algebras, Levitan's work on generalized translation operators, Brauer's work on pseudogroups, Hecke algebras, hypercomplex systems (referring to Berezansky and Kalyushnyi, Vainermann), paragroups (Ocneanu), superselection sectors (Doplicher, Haag and Roberts, Longo), Bose Mesner algebras, Racah Wigner algebras, centralizer algebras, table algebras (Arad and Blau), association schemes and the fusion rules of conformal field theories (Verlinde, Moore and Seiberg). You can look at the article for references.

Association schemes are for example hypergroups having renormalizations that can be realized by $0,1$-matrices and are very important in algebraic combinatorics and coding theory.