Hypergroup

This is a variation of group|Find other variations of group | Read a survey article on varying group

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

NOTE: This page is about the abstract algebraic notion of hypergroup. There is a related notion of hypergroup that comes up in probability theory and measure theory, which is at hypergroup (measure theory).

Definition

A hypergroup is a set $G$ equipped with a binary operation $*:G\times G\to 2^{G}\setminus \{\{\}\}$, i.e., a multi-valued binary operation, satisfying some conditions. The right side denotes the power set of $G$ minus the empty subset of $G$, because the binary operation is required to give at least one output for every input pair.

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

Examples

We can take the quotient of a hypergroup by any equivalence relation and get a hypergroup structure on the quotient set. Since groups are hypergroups to begin with, quotients of groups by various sorts of equivalence relations give hypergroups. Some related examples are:

• Conjugacy class hypergroup of a group.
• Character hypergroup of a group.