Book/article recommendations for an introduction to hypergroups and subsequent research

Question

I'm a grad student and I'm choosing an area to follow on my doctorate (in?) and I've been thinking about extension of topological group theory results to topological hypergroups, but for that I need to be introduced to the subject.

What is an awesome book which could serve as an introduction to the theory of hypergroups in general? As corollaries: is there such a book? Is the subject too new?

Bonus question: What are the directions this area is following/heading towards? Specific examples as papers would be more than welcome.

Answer 1

This paper seems to have the most citations among those I saw: Hypergroups and hypergroup algebras:http://link.springer.com/article/10.1007/BF01088201

But this book has even more (and some of it is on Google books): Generalized wavelets and hypergroups:http://www.amazon.com/Generalized-Wavelets-Hypergroups-Khalifa-Trimeche/dp/9056990802

I would recommend trying them out before buying (check out previews!) The level may be wrong.

Answer 2

See references in http://planetmath.org/hypergroup

Besides see B.Davvaz, W.Dudek, T.Vougiouklis, A generalization of $n$-ary algebraic systems. Communications in Algebra, v.37(2009), pp.1248–1263,

and other papers of W.Dudek.

Answer 3

The first thing to realize is that the term "hypergroup" is used for several different mathematical objects. Besides the generalisation of a group that you mention which allows the multiplication to be multivalued, there is also the notion of a hypergroup that are topological spaces with a convolution product for measures on this space which satisfies several axioms which are automatically satisfied for the convolution on a locally compact topological group. This direction comes from harmonic analysis, the reference given by Rushton concern this notion of hypergroup. See also the book by Bloom and Heyer http://books.google.fr/books?id=EUVGq4ciuMsC&dq=bloom+heyer&source=gbs_navlinks_s.

For hypergroups in your sense (hyperstructures) see also the Wikipedia article http://en.wikipedia.org/wiki/Hypergroup and its references.