Hypergroup

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QUICK PHRASES: variation of group where the multiplication operation is multi-valued

Definition

A hypergroup is a set $G$ equipped with a binary operation $*:G\times G\to 2^{G}$, i.e., a multi-valued binary operation, satisfying some conditions. Prior to stating the condition, we note that $*$ can be extended to an operation $2^{G}\times 2^{G}\to 2^{G}$ given by $X*Y=\bigcup _{x\in X,y\in Y}(x*y)$. Similarly, we can extend $*$ to operations $2^{G}\times G\to 2^{G}$ and $G\times 2^{G}\to 2^{G}$.

Condition name	What it means	Explanation
multi-valued version of associativity	For any $a,b,c\in G$, $(a*b)*c=a*(b*c)$ as sets.	Left side: Note that $a*b$ is a subset, say $S$, of $G$. $(a*b)*c=S*c$ is defined as the union $\bigcup _{s\in S}s*c$. Right side: Suppose $T=b*c$. Then, $a*(b*c)=a*T=\bigcup _{t\in T}(a*t)$.
multi-valued version of quasigroup type condition	For any $a\in G$, $a*G=G*a=G$.

Note that if the operation is single-valued and the underlying set of $G$ is non-empty, then $G$ becomes a group under $*$. This follows from the proof of associative quasigroup implies group (our statement is actually a little more general than that statement, because we are not assuming unique solutions to equations, but the proof does not use uniqueness).

Related notions

• Multiary group