Hypergroup

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 $$G \times 2^G \to 2^G$$.

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.