Hyperstructures

Hyperstructures are algebraic structures with a multi-valued operation. Hypergroups are the most prevalent in the literature, but there are also hypermonoids , hyperrings , hyperfields, and so on.

For motivation, David Corfield asks on MathOverflow: what are hypergroups and hyperrings good for? Further discussion at the n-Category Cafe .

Literature

TODO: General references

There is also an “analytic” version of a hypergroup: a measurable kernel \(X \times X \to \mathcal{M}(X)\) satisfying the group axioms.

  • Dunkl, 1973: The measure algebra of a locally compact hypergroup (doi, pdf)
  • Jewett, 1975: Spaces with an abstract convolution of measures (doi)
  • Bloom & Herbert, 1995: Harmonic analysis of probability measures on hypergroups

Recent papers on hyperfields (TODO: Connes)

  • Viro, 2010: Hyperfields for tropical geometry I. Hyperfields and dequantization (arxiv)
  • Rowen, 2016: Hypergroups and hyperfields in universal algebra (arxiv)