Strong hypergroups of order three

Abstract

This paper investigates the question of when a finite hypergroup with three elements is strong, that is satisfies the condition that its dual signed hypergroup is actually a hypergroup. We classify hermitian hypergroups of order three by weight into two dimensional families and show that the algebraic conditions arising from duality yield four interesting curves in the plane which bound the character values of strong and non strong hypergroups. By analysing the relations between these curves we discover that the stratum of strong hypergroups is connected for all weights in the range [4, ∞) except for the subinterval [5, 5 1 16 ] where there are two components. For weight equal to 5 the second component degenerates to a single point, the Golden hypergroup.

Non-commutative hypergroup of order five

“…From this fact, all hypergroups of order less than three are commutative. As was shown in [19], all hypergroups of order three are commutative. We next prove that all hypergroups of order four are commutative.…”

“…The structure of hypergroups has been studied by many authors (see [1-16, 18, 19] and references therein). N. J. Wildberger [19] determined the structure of hypergroups of order three; however the structure of hypergroups of order greater than three has not been determined. R. Ichihara and S. Kawakami [6] revealed the structure of hypergroups of order four which has a subhypergroup.…”

The adjacency matrices and the transition matrices related to random walks on graphs

“…Remark 4.3. According to [12,Section 4], the pre-hypergroup H = {c 0 , c 1 , c 2 } with the structure…”

Hypergroups and distance distributions of random walks on graphs

“…As was the case with groups, we can completely determine structures of hypergroups of low order. Structures of finite hypergroups of low orders have been studied in [18] and [15] for examples. In this paper, we shall treat hermitian discrete hypergroups which are generalizations of Z/2Z.…”