nLab hypermagma

Contents

Context

Algebra

Idea
Definition
Remarks
Example
Related entries
Reference

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 “multi-valued hyperalgebra” 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 $X$ can’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)).