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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).


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


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: