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Journal articles on the topic 'Hypergroups'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 September 2023

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1

Rösler, Margit, and Michael Voit. "Partial Characters and Signed Quotient Hypergroups." Canadian Journal of Mathematics 51, no. 1 (February 1, 1999): 96–116. http://dx.doi.org/10.4153/cjm-1999-006-6.

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AbstractIfGis a closed subgroup of a commutative hypergroupK, then the coset spaceK/Gcarries a quotient hypergroup structure. In this paper, we study related convolution structures onK/Gcoming fromdeformations of the quotient hypergroup structure by certain functions onKwhich we call partial characters with respect toG. They are usually not probability-preserving, but lead to so-called signed hypergroups onK/G. A first example is provided by the Laguerre convolution on [0, ∞[, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair (U(n, 1),U(n)) are discussed.

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2

Muruganandam, Varadharajan. "Fourier algebra of a hypergroup. I." Journal of the Australian Mathematical Society 82, no. 1 (February 2007): 59–83. http://dx.doi.org/10.1017/s144678870001747x.

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AbstractIn this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.

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3

Zhang, Xiaohong, Florentin Smarandache, and Yingcang Ma. "Symmetry in Hyperstructure: Neutrosophic Extended Triplet Semihypergroups and Regular Hypergroups." Symmetry 11, no. 10 (October 1, 2019): 1217. http://dx.doi.org/10.3390/sym11101217.

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The symmetry of hyperoperation is expressed by hypergroup, more extensive hyperalgebraic structures than hypergroups are studied in this paper. The new concepts of neutrosophic extended triplet semihypergroup (NET- semihypergroup) and neutrosophic extended triplet hypergroup (NET-hypergroup) are firstly introduced, some basic properties are obtained, and the relationships among NET- semihypergroups, regular semihypergroups, NET-hypergroups and regular hypergroups are systematically are investigated. Moreover, pure NET-semihypergroup and pure NET-hypergroup are investigated, and a strucuture theorem of commutative pure NET-semihypergroup is established. Finally, a new notion of weak commutative NET-semihypergroup is proposed, some important examples are obtained by software MATLAB, and the following important result is proved: every pure and weak commutative NET-semihypergroup is a disjoint union of some regular hypergroups which are its subhypergroups.

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4

De Salvo, Mario, Dario Fasino, Domenico Freni, and Giovanni Lo Faro. "G-Hypergroups: Hypergroups with a Group-Isomorphic Heart." Mathematics 10, no. 2 (January 13, 2022): 240. http://dx.doi.org/10.3390/math10020240.

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Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms.

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5

Kankaras, Milica, and Irina Cristea. "Fuzzy Reduced Hypergroups." Mathematics 8, no. 2 (February 17, 2020): 263. http://dx.doi.org/10.3390/math8020263.

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The fuzzyfication of hypercompositional structures has developed in several directions. In this note we follow one direction and extend the classical concept of reducibility in hypergroups to the fuzzy case. In particular we define and study the fuzzy reduced hypergroups. New fundamental relations are defined on a crisp hypergroup endowed with a fuzzy set, that lead to the concept of fuzzy reduced hypergroup. This is a hypergroup in which the equivalence class of any element, with respect to a determined fuzzy set, is a singleton. The most well known fuzzy set considered on a hypergroup is the grade fuzzy set, used for the study of the fuzzy grade of a hypergroup. Based on this, in the second part of the paper, we study the fuzzy reducibility of some particular classes of crisp hypergroups with respect to the grade fuzzy set.

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6

De Salvo, Mario, Dario Fasino, Domenico Freni, and Giovanni Lo Faro. "1-Hypergroups of Small Sizes." Mathematics 9, no. 2 (January 6, 2021): 108. http://dx.doi.org/10.3390/math9020108.

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In this paper, we show a new construction of hypergroups that, under appropriate conditions, are complete hypergroups or non-complete 1-hypergroups. Furthermore, we classify the 1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation β. Many of these hypergroups can be obtained using the aforesaid hypergroup construction.

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7

Gu, Ze. "On cyclic hypergroups." Journal of Algebra and Its Applications 18, no. 11 (August 19, 2019): 1950213. http://dx.doi.org/10.1142/s021949881950213x.

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In this paper, we introduce the concept of the index of a generator in a cyclic hypergroup, and show that a single power cyclic hypergroup is generated by an element with index [Formula: see text]. Also, a characterization of the fundamental relation on a cyclic hypergroup is given. Finally, we study corresponding quotient structures induced by regular (strongly regular) relations on cyclic hypergroups. As an application, the corresponding results on single power hypergroups are obtained.

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8

Kanwal, Shehzadi Salma, Naveed Yaqoob, Nabilah Abughazalah, and Muhammad Gulistan. "On Cyclic LA-Hypergroups." Symmetry 15, no. 9 (August 30, 2023): 1668. http://dx.doi.org/10.3390/sym15091668.

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Symmetries in the context of hypergroups and their generalizations are closely related to the algebraic structures and transformations that preserve certain properties of hypergroup operations. Symmetric LA-hypergroups are indeed commutative hypergroups. This paper considers a category of cyclic hyperstructures called the cyclic LA-semihypergroup that is a conception of LA-semihypergroups and cyclic hypergroups. We inaugurate the idea of cyclic LA-hypergroups. The interconnected notions of single-power cyclic LA-hypergroups, non-single power cyclic LA-hypergroups and some of their properties are explored.

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9

Al Tahan, M., and B. Davvaz. "On some properties of single power cyclic hypergroups and regular relations." Journal of Algebra and Its Applications 16, no. 11 (October 4, 2017): 1750214. http://dx.doi.org/10.1142/s0219498817502140.

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After introducing the notion of hypergroups by Marty, a number of generalizations of this fundamental concept has been studied. In this paper, we study a special type of hypergroups; single power cyclic hypergroups and present some of their properties. First, we determine the fundamental group of single power cyclic hypergroups. Next, we construct onto homomorphisms from any single power cyclic hypergroup to another defined hypergroups. Finally, we characterize all commutative single power cyclic hypergroups of order two.

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10

Leoreanu-Fotea, V., P. Corsini, A. Sonea, and D. Heidari. "Complete parts and subhypergroups in reversible regular hypergroups." Analele Universitatii "Ovidius" Constanta - Seria Matematica 30, no. 1 (February 1, 2022): 219–30. http://dx.doi.org/10.2478/auom-2022-0012.

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Abstract In this paper we analyse the center and centralizer of an element in the context of reversible regular hypergroups, in order to obtain the class equation in regular reversible hypergroups, by using complete parts. After an introduction in which basic notions and results of hypergroup theory are presented, particularly complete parts, then we give several properties, characterisations and also examples for the center and centralizer of an element for two classes of hypergroups. The next paragraph is dedicated to hypergroups associated with binary relations. We establish a connection between several types of equivalence relations, introduced by J. Jantosciak, such as the operational relation, the inseparability and the essential indistin-guishability and the conjugacy relation for complete hypergroups. Finally, we analyse Rosenberg hypergroup associated with a conjugacy relation.

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11

Voit, Michael. "Compact Almost Discrete Hypergroups." Canadian Journal of Mathematics 48, no. 1 (February 1, 1996): 210–24. http://dx.doi.org/10.4153/cjm-1996-010-8.

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AbstractA compact hypergroup is called almost discrete if it is homeomorphic to the one-point-compactification of a countably infinite discrete set. If the group Up of all p-adic units acts multiplicatively on the p-adic integers, then the associated compact orbit hypergroup has this property. In this paper we start with an exact projective sequence of finite hypergroups and use successive substitution to construct a new surjective projective system of finite hypergroups whose limit is almost discrete. We prove that all compact almost discrete hypergroups appear in this way—up to isomorphism and up to a technical restriction. We also determine the duals of these hypergroups, and we present some examples coming from partitions of compact totally disconnected groups.

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12

Massouros, Ch G. "On the semi-sub-hypergroups of a hypergroup." International Journal of Mathematics and Mathematical Sciences 14, no. 2 (1991): 293–304. http://dx.doi.org/10.1155/s0161171291000340.

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In this paper we study some properties of the semi-sub-hypergroups and the closed sub-hypergroups of the hypergroups. We introduce the correlated elements and the fundamental elements and we connect the concept antipodal of the latter with Frattin's hypergroup. We also present Helly's Theorem as a corollary of a more general Theorem.

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13

Massouros, Christos, and Gerasimos Massouros. "An Overview of the Foundations of the Hypergroup Theory." Mathematics 9, no. 9 (April 30, 2021): 1014. http://dx.doi.org/10.3390/math9091014.

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This paper is written in the framework of the Special Issue of Mathematics entitled “Hypercompositional Algebra and Applications”, and focuses on the presentation of the essential principles of the hypergroup, which is the prominent structure of hypercompositional algebra. In the beginning, it reveals the structural relation between two fundamental entities of abstract algebra, the group and the hypergroup. Next, it presents the several types of hypergroups, which derive from the enrichment of the hypergroup with additional axioms besides the ones it was initially equipped with, along with their fundamental properties. Furthermore, it analyzes and studies the various subhypergroups that can be defined in hypergroups in combination with their ability to decompose the hypergroups into cosets. The exploration of this far-reaching concept highlights the particularity of the hypergroup theory versus the abstract group theory, and demonstrates the different techniques and special tools that must be developed in order to achieve results on hypercompositional algebra.

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14

Voit, Michael. "Factorization of probability measures on symmetric hypergroups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 50, no. 3 (June 1991): 417–67. http://dx.doi.org/10.1017/s1446788700033012.

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AbstractGeneralizing known results for special examples, we derive a Khintchine type decomposition of probability measures on symmetric hypergroups. This result is based on a triangular central limit theorem and a discussion of conditions ensuring that the set of all factors of a probability measure is weakly compact. By our main result, a probability measure satisfying certain restrictions can be written as a product of indecomposable factors and a factor in I0(K), the set of all measures having decomposable factors only. Some contributions to the classification of I0(K) are given for general symmetric hypergroups and applied to several families of examples like finite symmetric hypergroups and hypergroup joins. Furthermore, all results are discussed in detail for a class of discrete symmetric hypergroups which are generated by infinitely many joins, for a class of countable compact hypergroups, for Sturm-Liouville hypergroups on [0, ∞[ and, finally, for polynomial hypergroups.

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15

Matsuzawa, Yasumichi, Hiromichi Ohno, Akito Suzuki, Tatsuya Tsurii, and Satoe Yamanaka. "Non-commutative hypergroup of order five." Journal of Algebra and Its Applications 16, no. 07 (June 30, 2016): 1750127. http://dx.doi.org/10.1142/s0219498817501274.

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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.

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16

De Salvo, Mario, Dario Fasino, Domenico Freni, and Giovanni Lo Faro. "On Hypergroups with a β-Class of Finite Height." Symmetry 12, no. 1 (January 15, 2020): 168. http://dx.doi.org/10.3390/sym12010168.

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In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms.

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17

Litvinov, G. L. "Hypergroups and hypergroup algebras." Journal of Soviet Mathematics 38, no. 2 (July 1987): 1734–61. http://dx.doi.org/10.1007/bf01088201.

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18

VOIT, MICHAEL. "CONTINUOUS ASSOCIATION SCHEMES AND HYPERGROUPS." Journal of the Australian Mathematical Society 106, no. 03 (July 27, 2018): 361–426. http://dx.doi.org/10.1017/s1446788718000149.

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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.

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19

Fechner, �ywilla, Eszter Gselmann, and L�szl� Sz�kelyhidi. "Generalized derivations and generalized exponential monomials on hypergroups." Opuscula Mathematica 43, no. 4 (2023): 493–505. http://dx.doi.org/10.7494/opmath.2023.43.4.493.

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In one of our former papers "Endomorphisms of the measure algebra of commutative hypergroups" we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra.

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20

Obata, Nobuaki, and Norman J. Wildberger. "Generalized hypergroups and orthogonal polynomials." Nagoya Mathematical Journal 142 (June 1996): 67–93. http://dx.doi.org/10.1017/s002776300000564x.

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We study in this paper a generalization of the notion of a discrete hypergroup with particular emphasis on the relation with systems of orthogonal polynomials. The concept of a locally compact hypergroup was introduced by Dunkl [8], Jewett [12] and Spector [25]. It generalizes convolution algebras of measures associated to groups as well as linearization formulae of classical families of orthogonal polynomials, and many results of harmonic analysis on locally compact abelian groups can be carried over to the case of commutative hypergroups; see Heyer [11], Litvinov [17], Ross [22], and references cited therein. Orthogonal polynomials have been studied in terms of hypergroups by Lasser [15] and Voit [31], see also the works of Connett and Schwartz [6] and Schwartz [23] where a similar spirit is observed.

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21

Willson, Benjamin. "A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability." Canadian Mathematical Bulletin 58, no. 2 (June 1, 2015): 415–22. http://dx.doi.org/10.4153/cmb-2014-069-3.

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AbstractIn this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

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22

Chvalina, Jan, and Šárka Hošková-Mayerová. "On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear First-Order Partial Differential Operators." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 1 (December 10, 2014): 85–103. http://dx.doi.org/10.2478/auom-2014-0008.

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AbstractThe contribution aims to create hypergroups of linear first-order partial differential operators with proximities, one of which creates a tolerance semigroup on the power set of the mentioned differential operators. Constructions of investigated hypergroups are based on the so called “Ends-Lemma” applied on ordered groups of differnetial operators. Moreover, there is also obtained a regularly preordered transpositions hypergroup of considered partial differntial operators.

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23

Kankaras, Milica. "Reducibility in Corsini hypergroups." Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, no. 1 (March 1, 2021): 93–109. http://dx.doi.org/10.2478/auom-2021-0007.

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Abstract In this paper, we study the reducibility property of special hyper-groups, called Corsini hypergroups, named after the mathematician who introduced them. The concept of reducibility was introduced by Jantosciak, who noticed that it can happen that hyperproduct does not distinguish between a pair of elements. He defined a certain equivalences in order to identify elements which play the same role with respect to the hyperoperation. First we will determine specific conditions under which the Corsini hypergroups are reduced. Next, we will present some properties of these hypergroups necessary for studying the fuzzy reducibility property. The fuzzy reducibility will be considered with respect to the grade fuzzy set μ̃, used for defining the fuzzy grade of a hypergroup. Finally, we will study the reducibility and the fuzzy reducibility of the direct product of Corsini hypergroups.

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24

Lindlbauer, Marc, and Michael Voit. "Limit theorems for isotropic random walks on triangle buildings." Journal of the Australian Mathematical Society 73, no. 3 (December 2002): 301–34. http://dx.doi.org/10.1017/s1446788700008995.

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AbstractThe spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

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25

Hu, Minghao, Florentin Smarandache, and Xiaohong Zhang. "On Neutrosophic Extended Triplet LA-hypergroups and Strong Pure LA-semihypergroups." Symmetry 12, no. 1 (January 14, 2020): 163. http://dx.doi.org/10.3390/sym12010163.

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We introduce the notions of neutrosophic extended triplet LA-semihypergroup, neutrosophic extended triplet LA-hypergroup, which can reflect some symmetry of hyperoperation and discuss the relationships among them and regular LA-semihypergroups, LA-hypergroups, regular LA-hypergroups. In particular, we introduce the notion of strong pure neutrosophic extended triplet LA-semihypergroup, get some special properties of it and prove the construction theorem about it under the condition of asymmetry. The examples in this paper are all from Python programs.

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26

Sonea, Andromeda Cristina, and Irina Cristea. "The Class Equation and the Commutativity Degree for Complete Hypergroups." Mathematics 8, no. 12 (December 21, 2020): 2253. http://dx.doi.org/10.3390/math8122253.

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The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.

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27

Nikkhah, A., B. Davvaz, and S. Mirvakili. "Hypergroups constructed from hypergraphs." Filomat 32, no. 10 (2018): 3487–94. http://dx.doi.org/10.2298/fil1810487n.

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The purpose of this paper is the study of hypergroups associated with hypergraphs. In this regard, we construct a hypergroupoid by defining a hyperoperation on the set of degrees of vertices of a hypergraph. We will see that the constructed hypergroupoid is always an Hv-group. We will investigate some conditions to have a hypergroup.

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Selvachandran, Ganeshsree, and Abdul Razak Salleh. "Vague Soft Hypergroups and Vague Soft Hypergroup Homomorphism." Advances in Fuzzy Systems 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/758637.

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We introduce and develop the initial theory of vague soft hyperalgebra by introducing the novel concept of vague soft hypergroups, vague soft subhypergroups, and vague soft hypergroup homomorphism. The properties and structural characteristics of these concepts are also studied and discussed.

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29

AL-TAHAN, MADELEINE, and BIJAN DAVVAZ. "Hypergroups Defined on Hypergraphs and their Regular Relations." Kragujevac Journal of Mathematics 46, no. 3 (June 2022): 487–98. http://dx.doi.org/10.46793/kgjmat2203.487t.

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The notion of hypergraphs, introduced around 1960, is a generalization of that of graphs and one of the initial concerns was to extend some classical results of graph theory. In this paper, we present some connections between hypergraph theory and hypergroup theory. In this regard, we construct two hypergroupoids by deﬁning two new hyperoperations on ℍ, the set of all hypergraphs. We prove that our deﬁned hypergroupoids are commutative hypergroups and we deﬁne hyperrings on ℍ by using the two deﬁned hyperoperations. Moreover, we study the fundamental group, complete parts, automorphism group and strongly regular relations of one of our hypergroups.

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30

Sonea, Andromeda, and Irina Cristea. "Euler's totient function applied to complete hypergroups." AIMS Mathematics 8, no. 4 (2023): 7731–46. http://dx.doi.org/10.3934/math.2023388.

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<abstract><p>We study the Euler's totient function (called also the Euler's phi function) in the framework of finite complete hypergroups. These are algebraic hypercompositional structures constructed with the help of groups, and endowed with a multivalued operation, called hyperoperation. On them the Euler's phi function is multiplicative and not injective. In the second part of the article we find a relationship between the subhypergroups of a complete hypergroup and the subgroups of the group involved in the construction of the considered complete hypergroup. As sample application of this connection, we state a formula that relates the Euler's totient function defined on a complete hypergroup to the same function applied to its subhypergroups.</p></abstract>

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31

Blasio, Bianca Di. "Hypergroups associated to harmonic NA groups." Journal of the Australian Mathematical Society 72, no. 2 (April 2002): 209–16. http://dx.doi.org/10.1017/s1446788700003852.

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AbstractA harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups.

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32

Bloom, Walter R., and Paul Ressel. "Positive Definite and Related Functions on Hypergroups." Canadian Journal of Mathematics 43, no. 2 (April 1, 1991): 242–54. http://dx.doi.org/10.4153/cjm-1991-013-2.

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AbstractIn this paper we make use of semigroup methods on the space of compactly supported probability measures to obtain a complete Lévy-Khinchin representation for negative definite functions on a commutative hypergroup. In addition we obtain representation theorems for completely monotone and completely alternating functions. The techniques employed here also lead to considerable simplification of the proofs of known results on positive definite and negative definite functions on hypergroups.

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Sahami, Amir, Mehdi Rostami, Seyedeh Fatemeh Shariati, and Salman Babayi. "On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces." Journal of Mathematics 2022 (January 29, 2022): 1–5. http://dx.doi.org/10.1155/2022/4939971.

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In this paper, we study the notion of approximate biprojectivity and left φ -biprojectivity of some Banach algebras, where φ is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra L 1 K implies that K is compact. Moreover, we investigate left φ -biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra B is left φ -biprojective if and only if K is compact, where K is a hypergroup. We also study the approximate biflatness and left φ -biflatness of hypergroup algebras in terms of amenability of their related hypergroups.

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34

Vati, Kedumetse, and László Székelyhidi. "Moment functions on hypergroup joins." Advances in Pure and Applied Mathematics 10, no. 3 (July 1, 2019): 215–20. http://dx.doi.org/10.1515/apam-2018-0027.

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Abstract Moment functions play a basic role in probability theory. A natural generalization can be defined on hypergroups which leads to the concept of generalized moment function sequences. In a former paper we studied some function classes on hypergroup joins which play a basic role in spectral synthesis. Moment functions are also important basic blocks of spectral synthesis. All these functions can be characterized by well-known functional equations. In this paper we describe generalized moment function sequences on hypergroup joins.

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35

Bloom, Walter R., and Herbert Heyer. "Non-symmetric translation invariant Dirichlet forms on hypergroups." Bulletin of the Australian Mathematical Society 36, no. 1 (August 1987): 61–72. http://dx.doi.org/10.1017/s0004972700026307.

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In this note translation-invariant Dirichlet forms on a commutative hypergroup are studied. The main theorem gives a characterisation of an invariant Dirichlet form in terms of the negative definite function associated with it. As an illustration constructions of potentials arising from invariant Dirichlet forms are given. The examples of one- and two-dimensional Jacobi hypergroups yield specifications of invariant Dirichlet forms, particularly in the case of Gelfand pairs of compact type.

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36

Medghalchi, A. R., and S. M. Tabatabaie. "An Extension of the Spectral Mapping Theorem." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–8. http://dx.doi.org/10.1155/2008/531424.

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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 .

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37

Muruganandam, Varadharajan. "Fourier algebra of a hypergroup - II. Spherical hypergroups." Mathematische Nachrichten 281, no. 11 (October 10, 2008): 1590–603. http://dx.doi.org/10.1002/mana.200510699.

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Iranmanesh, A., and A. H. Babareza. "Transposition hypergroups and complement hypergroups." Journal of Discrete Mathematical Sciences and Cryptography 6, no. 2-3 (January 2003): 161–68. http://dx.doi.org/10.1080/09720529.2003.10697973.

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Migliorato, Renato. "Canonicalv-hypergroups and ω-hypergroups." Journal of Discrete Mathematical Sciences and Cryptography 6, no. 2-3 (January 2003): 245–56. http://dx.doi.org/10.1080/09720529.2003.10697981.

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Lasser, Rupert, Josef Obermaier, and Holger Rauhut. "Generalized hypergroups and orthogonal polynomials." Journal of the Australian Mathematical Society 82, no. 3 (June 2007): 369–94. http://dx.doi.org/10.1017/s144678870003617x.

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AbstractThe concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

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Hoskova-Mayerova, Sarka, Madeline Al Tahan, and Bijan Davvaz. "Fuzzy Multi-Hypergroups." Mathematics 8, no. 2 (February 14, 2020): 244. http://dx.doi.org/10.3390/math8020244.

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A fuzzy multiset is a generalization of a fuzzy set. This paper aims to combine the innovative notion of fuzzy multisets and hypergroups. In particular, we use fuzzy multisets to introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroups. Different operations on fuzzy multi-hypergroups are defined and discussed and some results known for fuzzy hypergroups are generalized to fuzzy multi-hypergroups.

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ALAGHMANDAN, MAHMOOD, and MASSOUD AMINI. "DUAL SPACE AND HYPERDIMENSION OF COMPACT HYPERGROUPS." Glasgow Mathematical Journal 59, no. 2 (June 10, 2016): 421–35. http://dx.doi.org/10.1017/s0017089516000252.

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AbstractWe characterize dual spaces and compute hyperdimensions of irreducible representations for two classes of compact hypergroups namely conjugacy classes of compact groups and compact hypergroups constructed by joining compact and finite hypergroups. Also, studying the representation theory of finite hypergroups, we highlight some interesting differences and similarities between the representation theories of finite hypergroups and finite groups. Finally, we compute the Heisenberg inequality for compact hypergroups.

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Massouros, Gerasimos, and Christos Massouros. "Hypercompositional Algebra, Computer Science and Geometry." Mathematics 8, no. 8 (August 11, 2020): 1338. http://dx.doi.org/10.3390/math8081338.

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The various branches of Mathematics are not separated between themselves. On the contrary, they interact and extend into each other’s sometimes seemingly different and unrelated areas and help them advance. In this sense, the Hypercompositional Algebra’s path has crossed, among others, with the paths of the theory of Formal Languages, Automata and Geometry. This paper presents the course of development from the hypergroup, as it was initially defined in 1934 by F. Marty to the hypergroups which are endowed with more axioms and allow the proof of Theorems and Propositions that generalize Kleen’s Theorem, determine the order and the grade of the states of an automaton, minimize it and describe its operation. The same hypergroups lie underneath Geometry and they produce results which give as Corollaries well known named Theorems in Geometry, like Helly’s Theorem, Kakutani’s Lemma, Stone’s Theorem, Radon’s Theorem, Caratheodory’s Theorem and Steinitz’s Theorem. This paper also highlights the close relationship between the hyperfields and the hypermodules to geometries, like projective geometries and spherical geometries.

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De Salvo, Mario De, Dario Fasino, Domenico Freni, and Giovanni Lo Lo Faro. "Commutativity and Completeness Degrees of Weakly Complete Hypergroups." Mathematics 10, no. 6 (March 18, 2022): 981. http://dx.doi.org/10.3390/math10060981.

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We introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the β-classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a suitably defined relation over G. The commutativity degree of weakly complete hypergroups can be related to that of the underlying group. Furthermore, in analogy to the degree of commutativity, we introduce the degree of completeness of finite hypergroups and analyze this degree for weakly complete hypergroups in terms of their β-classes.

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Leoreanu, Violeta. "Commutative hypergroups associated with arbitrary hypergroups." Journal of Discrete Mathematical Sciences and Cryptography 6, no. 2-3 (January 2003): 195–98. http://dx.doi.org/10.1080/09720529.2003.10697975.

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Leoreanu-Fotea, Violeta. "Approximations in hypergroups and fuzzy hypergroups." Computers & Mathematics with Applications 61, no. 9 (May 2011): 2734–41. http://dx.doi.org/10.1016/j.camwa.2011.03.030.

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Lang, W. Christopher. "The structure of hypergroup measure algebras." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 46, no. 2 (April 1989): 319–42. http://dx.doi.org/10.1017/s1446788700030809.

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AbstractA close analogue for some hypergroup measure algebras of the structure semigroup theorem of J. L. Taylor for convolution measure algebras is constructed: a structure semihypergroup representation is made for the hypergroup measure and its spectrum. This is done for those hypergroup measure algebras that satisfy a condition known as the structure-strong condition. This condition is that the norm-closure of the linear span of the spectrum of the hypergroup measure algebra is a commutative B*-algebra. Then examples of hypergroups whose measure algebras satisfy this condition are given. They include the space of B-orbits of G, where B is a finite solvable group of automorphisms on a locally compact abelian group G. (The hypergroup measure algebra may be identified with the algebra of B-invariant measures on G.) Other examples are the algebra of central measures on a compact, connected, semisimple Lie group, and the algebra of rotation invariant measures on the plane.

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RENTZSCH, CHRISTIAN. "LÉVY–KHINTCHINE REPRESENTATION ON LOCAL STURM–LIOUVILLE HYPERGROUPS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 02, no. 01 (March 1999): 79–104. http://dx.doi.org/10.1142/s0219025799000059.

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We introduce local Sturm–Liouville hypergroups and prove a Lévy–Khintchine formula for generators of convolution semigroups on these hypergroups. Then we show that many well-known hypergroups are local Sturm–Liouville hypergroups. Moreover, we study the canonical form of the generators on the given examples.

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Massouros, Christos G., and Naveed Yaqoob. "On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations." Mathematics 9, no. 15 (August 3, 2021): 1828. http://dx.doi.org/10.3390/math9151828.

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This paper presents the study of algebraic structures equipped with the inverted associativity axiom. Initially, the definition of the left and the right almost-groups is introduced and afterwards, the study is focused on the more general structures, which are the left and the right almost-hypergroups and on their enumeration in the cases of order 2 and 3. The outcomes of these enumerations compared with the corresponding in the hypergroups reveal interesting results. Next, fundamental properties of the left and right almost-hypergroups are proved. Subsequently, the almost hypergroups are enriched with more axioms, like the transposition axiom and the weak commutativity. This creates new hypercompositional structures, such as the transposition left/right almost-hypergroups, the left/right almost commutative hypergroups, the join left/right almost hypergroups, etc. The algebraic properties of these new structures are analyzed and studied as well. Especially, the existence of neutral elements leads to the separation of their elements into attractive and non-attractive ones. If the existence of the neutral element is accompanied with the existence of symmetric elements as well, then the fortified transposition left/right almost-hypergroups and the transposition polysymmetrical left/right almost-hypergroups come into being.

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Wang, Shuanhong. "Algebraic quantum hypergroups of discrete type." MATHEMATICA SCANDINAVICA 108, no. 2 (June 1, 2011): 198. http://dx.doi.org/10.7146/math.scand.a-15167.

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In this paper we will study some structures of algebraic quantum hypergroups. First, we construct more examples of algebraic quantum hypergroups of discrete type. Next, we introduce the notion of a generalized quasi-Frobenius multiplier Hopf algebra and then show that generalized quasi-Frobenius multiplier Hopf algebras are a class of algebraic quantum hypergroups of discrete type. We also give some equivalent conditions for an algebraic quantum group to be of discrete type. Finally, we study sub-algebraic quantum hypergroups of discrete type and quotients of algebraic quantum hypergroups of discrete type.

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