Moment functions and exponential monomials on commutative hypergroups

The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.

Fundamental relations on hyper bi-modules and characterizations of complete hyper bi-modules

A hyper bi-module is a commutative hypergroup that is both a left and a right hypermodule, such that the left and the right multiplications are compatible. We define the fundamental relation on an [Formula: see text]-hyper bi-module, where [Formula: see text] and [Formula: see text] are hyperrings and the left and the right multiplications are compatible. Also, we state some conditions that are equivalent to the transitivity of this relation and finally we characterize the complete [Formula: see text]-hyper bi-modules.

Weighted Orlicz algebras on hypergroups

Let K be a hypergroup, w be a weight function and let (?,?) be a complementary pair of Young functions. We consider the weighted Orlicz space L??(K) and investigate some of its algebraic properties under convolution. We also study the existence of an approximate identity for the Banach algebra L?w(K). Further, we describe the maximal ideal space of the convolution algebra L?w(K) for a commutative hypergroup K.

CONTINUOUS ASSOCIATION SCHEMES AND HYPERGROUPS

Classical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated with discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$ . We study some basic results for this notion and present several classes of examples. It turns out that, for a given commutative hypergroup, the existence of a related continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We, in particular, show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.

Commutative hypergroups associated with a hyperfield

Let [Formula: see text] be a commutative hypergroup and [Formula: see text] a discrete commutative hypergroup. In this paper we introduce a commutative hypergroup [Formula: see text] associated with a hyperfield [Formula: see text] of [Formula: see text] based on [Formula: see text]. Moreover, for the hyperfield [Formula: see text] of a compact commutative hypergroup [Formula: see text] of strong type based on a discrete commutative hypergroup [Formula: see text] of strong type, we introduce the dual hyperfield [Formula: see text] of [Formula: see text] based on [Formula: see text] and show that [Formula: see text].

On the g-hypergroupoids associated with g-hypergraphs

In this paper, we associate a partial g-hypergroupoid with a given g-hypergraph and analyze the properties of this hyperstructure. We prove that a g-hypergroupoid may be a commutative hypergroup without being a join space. Next, we define diagonal direct product of g-hypergroupoids. Further, we construct a sequence of g-hypergroupoids and investigate some relationships between it?s terms. Also, we study the quotient of a g-hypergroupoid by defining a regular relation. Finally, we describe fundamental relation of an Hv-semigroup as a g-hypergroupoid.

Non-commutative hypergroup of order five

We prove that all hypergroups of order four are commutative and that there exists a non-comutative hypergroup of order five. These facts imply that the minimum order of non-commutative hypergroups is five, even though the minimum order of non-commutative groups is six.