Hypergroup

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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

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: