nLab hypergroup




A hypergroup is a algebraic structure similar to a group, but where the composition operation does not just take two elements to a single product element in the group, but to a subset of elements of the group.

It is a hypermonoid with additional groupal structure and property.


A canonical hypergroup is a set, HH, equipped with a commutative binary operation,

+:H×H𝒫(H)\{} + : H \times H \to \mathcal{P}(H)\backslash\{\emptyset\}

whose value is a non-empty subset of HH, and a zero element 0H0 \in H, such that

  1. ++ is associative (extended to allow addition of subsets of HH);
  2. 0+x={x}=x+0,xH0 + x = {\{x\}} = x + 0, \forall x \in H;
  3. xH,!yH\forall x \in H, \exists ! y \in H such that 0x+y0 \in x + y (we denote this yy as x-x);
  4. x,y,zH,xy+z\forall x, y, z \in H, x \in y + z implies zxyz \in x - y (where xyx - y means x+(y)x + (-y) as usual).

𝒫(H)\mathcal{P}(H) is the notation for the power set of HH. A related variant is the notion of n-valued group.


The additive structure underlying a hyperring is a canonical hypergroup. See there for more examples.


See also at hypermagma and multivalued group.

  • J. Delsarte, Hypergroupes et opérateurs de permutation et de transmutation, La théorie des équations aux dérivées partielles. Nancy, 9-15 avril 1956, pp. 29–45 Colloq. Internat. CNRS, LXXI [International Colloquia of the CNRS] Centre National de la Recherche Scientifique, Paris, 1956 MR0116151

  • J. Jantosciak, Transposition hypergroups: Noncommutative join spaces , J. Algebra 187 (1997) pp.97-119.

  • G. L. Litvinov, Hypergroups and hypergroup algebras , arXiv:1109.6596

  • F. Marty, Sur une généralization de la notion de groupe , IV Congrès des Mathématiciens Scandinaves, Stockholm 1934.

category: algebra

Last revised on April 16, 2024 at 12:36:06. See the history of this page for a list of all contributions to it.