Hypergroup deformations of semigroups

  • Vishvesh KumarEmail author
  • Kenneth A. Ross
  • Ajit Iqbal Singh
Research Article


We view the well-known example of the dual of a countable compact hypergroup, motivated by the orbit space of p-adic integers by Dunkl and Ramirez (Trans Am Math Soc 202:339–356, 1975), as hypergroup deformation of the max semigroup structure on the linearly ordered set \(\mathbb {Z}_+\) of the non-negative integers along the diagonal. This works as motivation for us to study hypergroups or semi convolution spaces arising from “max” semigroups or general commutative semigroups via hypergroup deformation on idempotents.


Semigroups “Max” semigroups Discrete hypergroups Discrete semi-hypergroup Dunkl–Ramirez example Hypergroup deformation of idempotents Dual hypergroups 



The authors thank the referee for his/her kind comments and suggestions.

Vishvesh Kumar thanks the Council of Scientific and Industrial Research, India, for its senior research fellowship. He thanks his supervisors Ritumoni Sarma and N. Shravan Kumar for their support and encouragement. A preliminary version of a part of this paper was included in the invited talk by Ajit Iqbal Singh at the conference “The Stone-\(\check{\text{ C }}\)ech compactification: Theory and Applications, at Centre for Mathematical Sciences, University of Cambridge, July 6–8 2016” in honour of Neil Hindman and Dona Strauss. She is grateful to the organizers H.G. Dales and Imre Leader for the kind invitation, hospitality and travel support. She thanks them, Dona Strauss and Neil Hindman and other participants for useful discussion. She expresses her thanks to the Indian National Science Academy for the position of INSA Emeritus Scientist and travel support.


  1. 1.
    Alaghmandan, M., Samei, E.: Weighted discrete hypergroups. Indiana Univ. Math. J. 65(2), 423–451 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bloom, W. R., Heyer, H.: Harmonic Analysis on Probability Measures on Hypergroups. De Gruyter, Berlin (1995) (Reprint: 2011)Google Scholar
  3. 3.
    Berglund, J.F., Junghenn, H.D., Milnes, P.: Analysis on Semigroups: Function Spaces, Compactifications, Representations, Canadian Mathematical Society Series of Monographs and Advanced Texts). Wiley, New York (1989)Google Scholar
  4. 4.
    Chilana, A.K., Kumar, A.: Ultra-strong Ditkin sets in hypergroups. Proc. Am. Math. Soc. 77(3), 353–358 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Connett, W.C., Gebuhrer, O., Schwartz, A.L. (eds.): Application of Hypergroups and Related Measure Algebras, vol. 183. American Mathematical Society, Contemporary Mathematics, Providence, RI (1995)Google Scholar
  6. 6.
    Dunkl, C.F.: The measure algebra of a locally compact hypergroup. Trans. Am. Math. Soc. 179, 331–348 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dunkl, C.F., Ramirez, D.E.: A family of countable compact \(\text{ P }_\star \)-hypergroups. Trans. Am. Math. Soc. 202, 339–356 (1975)zbMATHGoogle Scholar
  8. 8.
    Hewitt, E., Zuckerman, H.S.: Structure theory for a class of convolution algebras. Pac. J. Math. 7(1), 913–941 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis Vol. I: Structure of Topological Groups, Integration Theory, Group Representations, 2nd edn., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer, Berlin (1979)Google Scholar
  10. 10.
    Hofmann, K.H., Lawson, J.D.: Linearly ordered semigroups: historical origins and A. H. Cliffords influence. In: Semigroup Theory and Its Applications (New Orleans, LA, 1994). London Mathematical Society. Lecture Note Series, vol. 231, Cambridge University Press, Cambridge, pp. 15–39 (1996)Google Scholar
  11. 11.
    Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18, 1–101 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kawakami, S., Tsurii, T., Yamanaka, S.: Deformations of finite hypergroups. Sci. Math. Jpn. 79(2), 213–223 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Petrich, M.: Semicharacters of the cartesian product of two semigroups. Pac. J. Math. 12(2), 679–683 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ross, K.A.: The structure of certain measure algebras. Pac. J. Math. 11(2), 723–737 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ross, K.A., Xu, D.: Hypergroup deformations and Markov chains. J. Theoret. Probab. 7(4), 813–830 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ross, K.A., Anderson, J.M., Litvinov, G.L., Singh, A.I., Sunder, V.S., Wildberger, N.J. (eds.): Harmonic Analysis and Hypergroups. Trends Math., Birkh\(\ddot{\text{ a }}\)user Boston, Boston, MA (1998)Google Scholar
  17. 17.
    Spector, R.: Apercu de la th\(\acute{\text{ e }}\)orie des hypergroupes. In: Analyse Harmonique sur les Groupes de Lie (1975) pp. 643–673 (Sem. Nancy-Strasburg 1973–1975, Lecture Notes in Mathematics 497, Springer, Berlin (1975)Google Scholar
  18. 18.
    Voit, M.: Compact almost discrete hypergroups. Canad. J. Math. 48, 210–224 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vrem, R.C.: Harmonic analysis on compact hypergroups. Pac. J. Math. 85(1), 239–251 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vrem, R.C.: Hypergroup joins and their dual objects. Pac. J. Math. 111(2), 483–495 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  • Vishvesh Kumar
    • 1
    Email author
  • Kenneth A. Ross
    • 2
  • Ajit Iqbal Singh
    • 3
  1. 1.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia
  2. 2.Department of MathematicsUniversity of OregonEugeneUSA
  3. 3.INSA Emeritus ScientistThe Indian National Science AcademyNew DelhiIndia

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