Contents

# Contents

## Idea

A hypermagma $X$ relates to a magma like a hypergraph relates to an ordinary graph i. e. the binary operation on $X$ becomes multi-valued by taking value in $2^X$ instead of $X$.

By imposing further axioms one obtains the concept of a hypergroup corresponding to the generalization of the group concept in this context.

With the commutative variant of a canonical hypergroup this βmulti-valued hyperalgebraβ has recently gained prominence in number theory and algebraic geometry over $\mathbb{F}_1$ (cf. hyperring).

## Definition

A set $X$ together with a function $X\times X\to 2^X$ is called a hypermagma. The function is normally denoted by $\cdot$ and called the hyperlaw (, hyperoperation or hyperproduct) of $X$. A morphism of hypermagmas from $X$ to $Y$ is a function $f:X\to Y$ such that $f(x\cdot y)\subset f(x)\cdot f(y)$ for all $x,y\in X$. The morphism is called good if $f(x\cdot y) = f(x)\cdot f(y)$.

A hypermagma $X$ that satisfies furthermore:

• $(x\cdot y)\cdot z = x\cdot (y\cdot z)$ for all $x,y,z\in X$ ( associativity ) and

• $x\cdot X= X = X\cdot x$ for all $x\in X$ ( reproduction )

is called a hypergroup.

## Remarks

Given a hypermagma $X$ the hyperlaw is extended to $A,B\in 2^X$ by $A \cdot B=\cup_{a\in A,b\in B} a\cdot b$. Hence, $x\cdot Y$ is understood as $\{x\}\cdot Y$ etc.

Suppose $X$ is an associative hypermagma and $x\cdot y=\emptyset$ then $x\cdot y \cdot X=\emptyset$ and, accordingly, $y\cdot X\neq\X$ or $x\cdot X\neq X$ whence $X$ canβt be a hypergroup. So we see that the hyperlaws of hypergroups are in fact valued in non-empty subsets - hypermagmas with this property are sometimes called hypergroupoids.

But note that $\emptyset$ together with the empty map is nevertheless a hypergroup, in fact, the initial hypergroup in the obvious category of hypergroups $\mathbf{HypGrp}$.

By imposing commutativity one arrives at the notion of a canonical hypergroup that enters into the definition of a hyperring.

Hypergroups whose hyperlaw is valued in singleton subsets correspond to groups.

## Example

Let $G$ be a compact Lie group and $\hat{G}$ the set of its irreducible representations. Given $a,b\in\hat{G}$ define $a\cdot b$ as the set of irreducible representations $\mu_1,\dots,\mu_k$ occurring in the decomposition $a\otimes b=\Sum m_i \mu_i$. This endows $\hat{G}$ with the structure of a hypergroup. (For further details and the connection to operator algebra see Litvinov (2011)).

## Reference

• S. D. Comer, Polygroups derived from cogroups , J. Algebra 89 no.2 (1984) 397β405 doi

• P. Corsini, V. Leoreanu-Fotea, Applications of yhperstructure theory , Kluwer Dordrecht 2003.

• L. Haddad, Y. Sureau, Les cogroupes et la construction de Utumi , Pacific J. Math. 145 no.1 (1990) 17β58. (abstract)

• L. Haddad, Y. Sureau, Les groupes, les hypergroupes et lβΓ©nigme des Murngin , JPAA 87 (1993) pp.221-235.

• Louis H. Rowen, Residue structures, arXiv:2403.09467

category: algebra

Last revised on April 16, 2024 at 12:28:54. See the history of this page for a list of all contributions to it.