Hypergroups are generalizationsPlanetmathPlanetmath of groups. Recall that a group is set with a binary operationMathworldPlanetmath on it satisfying a number of conditions. If this binary operation is taken to be multivalued, then we arrive at a hypergroup. In order to make this precise, we need some preliminary concepts:

Definition. A hypergroupoid, or multigroupoid, is a non-empty set G, together with a multivalued function :G×GG called the multiplication on G.

We write ab, or simply ab, instead of (a,b). Furthermore, if ab={c}, then we use the abbreviation ab=c.

A hypergroupoid is said to be commutativePlanetmathPlanetmathPlanetmath if ab=ba for all a,bG. Defining associativity of on G, however, is trickier:

Given a hypergroupoid G, the multiplication induces a binary operation (also written ) on P(G), the powerset of P, given by

AB:={abaA and bB}.

As a result, we have an induced groupoidPlanetmathPlanetmathPlanetmathPlanetmath P(G). Instead of writing {a}B, A{b}, and {a}{b}, we write aB,Ab, and ab instead. From now on, when we write (ab)c, we mean “first, take the productMathworldPlanetmath of a and b via the multivalued binary operation on G, then take the product of the set ab with the element c, under the induced binary operation on P(G)”. Given a hypergroupoid G, there are two types of associativity we may define:

Type 1.

(ab)ca(bc), and

Type 2.


G is said to be associative if it satisfies both types of associativity laws. An associative hypergroupoid is called a hypersemigroup. We are now ready to formally define a hypergroup.

Definition. A hypergroup is a hypersemigroup G such that aG=Ga=G for all aG.

For example, let G be a group and H a subgroupMathworldPlanetmathPlanetmath of G. Let M be the collectionMathworldPlanetmath of all left cosetsMathworldPlanetmath of H in G. For aH,bHM, set


Then M is a hypergroup with multiplication .

If the multiplication in a hypergroup G is single-valued, then G is a 2-group (http://planetmath.org/PolyadicSemigroup), and therefore a group (see proof here (http://planetmath.org/PolyadicSemigroup)).

Remark. A hypergroup is also known as a multigroup, although some call a multigroup as a hypergroup with a designated identity elementMathworldPlanetmath e, as well as a designated inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath for every element with respect e. Actually identities and inverses may be defined more generally for hypergroupoids:

Let G be a hypergroupoid. Identity elements are defined via the following three sets:

  1. 1.

    (set of left identities): IL(G):={eGaea for all aG},

  2. 2.

    (set of right identities): IR(G):={eGaae for all aG}, and

  3. 3.

    (set of identities): I(G)=IL(G)IR(G).

eL(G) is called an absolute identity if ea=ae=a for all aG. If e,fG are both absolute identities, then e=ef=f, so G can have at most one absolute identity.

Suppose eIL(G)IR(G) and aG. An element bG is said to be a left inverse of a with respect to e if eba. Right inverses of a are defined similarly. If b is both a left and a right inverse of a with respect to e, then b is called an inverse of a with respect to e.

Thus, one may say that a multigroup is a hypergroup G with an identity eG, and a function :-1GG such that a-1:=-1(a) is an inverse of a with respect to e.

In the example above, M is a multigroup in the sense given in the remark above. The designated identity is H (in fact, this is the only identity in M), and for every aHM, its designated inverse is provided by a-1H (of course, this may not be its only inverse, as any bH such that ahb=e for some hH will do).


  • 1 R. H. Bruck, A Survey on Binary Systems, Springer-Verlag, New York, 1966.
  • 2 M. Dresher, O. Ore, Theory of Multigroups, Amer. J. Math. vol. 60, pp. 705-733, 1938.
  • 3 J.E. Eaton, O. Ore, Remarks on Multigroups, Amer. J. Math. vol. 62, pp. 67-71, 1940.
  • 4 L. W. Griffiths, On Hypergroups, Multigroups, and Product Systems, Amer. J. Math. vol. 60, pp. 345-354, 1938.
  • 5 A. P. Dičman, On Multigroups whose Elements are Subsets of a Group, Moskov. Gos. Ped. Inst. Uč. Zap. vol. 71, pp. 71-79, 1953
Title hypergroup
Canonical name Hypergroup
Date of creation 2013-03-22 18:38:22
Last modified on 2013-03-22 18:38:22
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 20N20
Synonym multigroupoid
Synonym multisemigroup
Synonym multigroup
Related topic group
Defines hypergroupoid
Defines hypersemigroup
Defines left identity
Defines right identity
Defines identity
Defines absolute identity
Defines left inverse
Defines right inverse
Defines inverse
Defines absolute identity