Commutative Hypergroups Research Articles

Intuitionistic Fuzzy Sets And Join Spaces Associated With Ary Membership Functions

In this paper, we associate finite hyperstructures with fuzzy sets endowed with n-ary membership functions and analyze the properties of this new hyperstructures. We prove that the new hyperstructure is a commutative hypergroup, but generally it is not a join space. We give some conditions such that the hypergroup has this property. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hypergroups of a hypergroup.

Generalized fuzzy n-ary subhypergroups of a commutative n-ary hypergroup

Abstract In this paper, by means of a new idea, the concept of (invertible) (∈,∈∨q)-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are investigated. A kind of n-ary quotient hypergroup by an (∈, ∈∨q)-fuzzy n-ary subhypergroup is provided and the relationships among (∈, ∈∨q)-fuzzy n-ary subhypergroups, n-ary quotient hypergroups and homomorphism are investigated. Several isomorphism theories of n-ary hypergroups are established.

Strongly invariant means on commutative hypergroups

We introduce and study strongly invariant means $m$ on commutative hypergroups, $m(T_x\varphi \cdot \psi)=m(\varphi \cdot T_{\tilde{x}}\psi)$, $x \in K$, $\varphi,\psi \in L^\infty(K)$. We show that the existence of such means is equivalent to a strong Re

A moment problem on some types of hypergroups

‎The classical moment problem is formulated for commutative hypergroups and the uniqueness is proved for polynomial hypergroups in a single variable and for Sturm--Liouville hypergroups‎.

Spectral Analysis and Moment Functions

We show that on commutative hypergroups the existence of nonzero moment functions in a variety is closely related to spectral analysis for the variety.

Spectra ofL1-convolution operators acting onLp-spaces of commutative hypergroups

AbstractWe show that, for commutative hypergroups, the spectrum of allL1-convolution operators onLpis independent ofp∈ [1, ∞] exactly when the Plancherel measure is supported on the whole character space χb(K), i.e., exactly whenL1(K) is symmetric and for every α ∈Reiter's conditionP2holds true. Furthermore, we explicitly determine the spectra σp(Tϵ1) for the family of Karlin–McGregor polynomial hypergroups, which demonstrate that in general the spectra might even be different for eachp.

A cohomology approach to the extension problem for commutative hypergroups

The purpose of this paper is to determine all commutative hypergroup extensions of a countable discrete commutative hypergroup by a locally compact Abelian group, in terms of second order cohomology of hypergroups, a notion which generalizes the cohomology of groups.

Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups.

Decomposition of conditionally positive definite functions on commutative hypergroups

In this paper we consider bounded, conditionally positive definite functions on commutative hypergroups. We show an integral representation that allows to decompose conditionally positive definite functions as differences of positive definite functions.

Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

Let $p,q$ be positive integers. The groups $U_p(\mathbb{C})$ and $U_p(\mathbb{C})\times U_q(\mathbb{C}) $ act on the Heisenberg group $H_{p,q}:=M_{p,q}(\mathbb{C})\times \mathbb{R}$ canonically as groups of automorphisms, where $M_{p,q}(\mathbb{C})$ is

Fuzzy roughness of n-ary hypergroups based on a complete residuated lattice

This paper considers the relationships among L-fuzzy sets, rough sets and n-ary hypergroup theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy n-ary subhypergroup are introduced and studied. Then, a new algebraic structure called (invertible) L-fuzzy rough n-ary subhypergroups is defined, and the (strong) homomorphism of lower and upper L-fuzzy rough approximation operators is studied.

Spectral analysis on commutative hypergroups

The problem of spectral analysis is formulated on commutative hypergroups and is solved for finite dimensional varieties.

Martingale characterizations of increment processes in a commutative hypergroup

Martingale characterizations and the related martingale problem are studied for increments processes with values in a commutative hypergroup. For the proofs Fourier theory is applied for the convolution hemigroups associated with the increment processes.

α-Amenable hypergroups

Let K denote a locally compact commutative hypergroup, L1(K) the hypergroup algebra, and α a real-valued hermitian character of K. We show that K is α-amenable if and only if L1(K) is α-left amenable. We also consider the α-amenability of hypergroup joins and polynomial hypergroups in several variables as well as a single variable.

Lévy Khinchin Formula on Commutative Hypercomplex System

Abstract. A commutative hypercomplex system L 1 (Q,m) is, roughly speaking, a spacewhich is defined by a structure measure (c(A,B,r),(A,B ∈ β(Q)). Such space has beenstudied by Berezanskii and Krein. Our main purpose is to establish a generalization ofconvolution semigroups and to discuss the role of the L´evy measure in the L´evy-Khinchinrepresentation in terms of continuous negative definite functions on the dual hypercomplexsystem. 1. IntroductionThe integral representation of negative definite functions is known in the liter-ature as the L´evy-Khinchin formula. This was established for G= Rin the late1930’s by L´evy and Khinchin. It had been extended to Lie groups by Hunt [9] andby Parthasarathy et al [13] to locally compact abelian groups with a countable case.In 1969 Harzallah [7] gave a representation formula for an arbitrary locally compactabelian group. Hazod [8] obtained a L´evy-Khinchin formula for an arbitrary locallycompact group. The general L´evy-Khinchin formula and the special case, where theinvolution is identical are due to Berg [4]. Lasser [12] deduced the L´evy-Khinchinformula for commutative hypergroups. Now these contribution may be viewed asa L´evy-Khinchin formula for negative definite functions defined on commutativehypercomplex systems.Let Q be a complete separable locally compact metric space of pointsp,q,r··· ,β(Q) be the σ-algebra of Borel subsets, and β

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ℝ,ℂ,ℍ were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.

Hyperstructures and fuzzy sets endowed with two membership functions

This paper contains new results concerning finite hyperstructures associated with fuzzy sets endowed with two membership functions. In general, these hyperstructures are commutative hypergroups, but not join spaces. We give some conditions such that the considered hypergroups have also this property and we present examples of their fuzzy subhypergroups. Then we analyze two sequences of join spaces and fuzzy sets associated with a finite hypergroupoid, using fuzzy sets endowed with two membership functions.

The semisimplicity of L 1(K, w) of a weighted commutative hypergroup K

In the present paper, it is shown that, for a locally compact commutative hypergroup K with a Borel measurable weight function w, the Banach algebra L 1(K,w) is semisimple if and only if L 1(K) is semisimple. Indeed, we have improved a well-krown result of Bhatt and Dedania from locally compact groups to the general setting of locally compact hypergroups.

An Extension of the Spectral Mapping Theorem

We give an extension of the spectral mapping theorem on hypergroups and prove that if is a commutative strong hypergroup with and is a weakly continuous representation of on a -algebra such that for every , is an -automorphism, is a synthesis set for and is without order, then for any in a closed regular subalgebra of containing , , where is the Arveson spectrum of .

Bessel convolutions on matrix cones

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras ${\mathbb F} = {\mathbb R}, {\mathbb C}$ or ${\mathbb H}$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}({\mathbb F})$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p({\mathbb F})$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures depending only on the matrix spectra. Under the mapping $r\mapsto \text{spec}(r)$, the convolutions on the underlying matrix cone induce a continuous series of hypergroup convolutions on a Weyl chamber of type $B_q$. The characters are now Dunkl-type Bessel functions. These convolution algebras on the Weyl chamber naturally extend the harmonic analysis for Cartan motion groups associated with the Grassmann manifolds $U(p,q)/(U_p\times U_q)$ over ${\mathbb F}$.