Skip to main content

An Addendum to “Hypergroup deformations of semigroups”

The Original Article was published on 29 January 2019

Unfortunately, in [2] we overlooked a fundamental class of examples studied by Michael Voit ([3], Sections 5 and 6), which are also generalizations of the class of Dunkl-Ramirez discrete hypergroups [1], and their duals are also almost discrete, i.e., one-point compactifications of discrete countably infinite spaces. These examples are clearly (hermitian) hypergroup deformations of the semigroup \((\mathbb {Z}_+,<,\max )\).

Voit studied his class of examples to illustrate factorization of probability measures on certain symmetric, i.e., hermitian hypergroups. In [2], we arrived at the class via necessary conditions for a hypergroup deformation \((S,*)\) of an infinite “max” semigroup \((S, <, \cdot )\) with identity, the first necessary condition being that \((S, <, \cdot )\) is isomorphic to \((\mathbb {Z}_+,<,\max )\). The sufficiency had a simple computational proof. Of course, it should have been attributed to Voit [3], had we been aware of this class of examples. As is clear from our Theorem 3.2, general discussion after Remark 3 and Corollary 3.3, the two classes, viz., Voit’s class and the (hermitian) hypergroup deformations of \((\mathbb {Z}_+,<,\max )\), coincide. Because Voit discusses the dual \(\widehat{K}\) of any such hypergroup K, the detailed proofs in Sect. 3.3 should have been avoided.

The authors thank Michael Voit for relevant information.

References

  1. Dunkl, C.F., Ramirez, D.E.: A family of countable compact \(P_*\)-hypergroups. Trans. Am. Math. Soc. 202, 339–356 (1975)

    MATH  Google Scholar 

  2. Kumar, V., Ross, K.A., Singh, A.I.: Hypergroup deformations of semigroups. Semigroup Forum (2019). https://doi.org/10.1007/s00233-019-10003-6

    Google Scholar 

  3. Voit, M.: Factorization of probability measures on symmetric hypergroups. J. Aust. Math. Soc. (Ser. A) 50, 417–467 (1991)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vishvesh Kumar.

Additional information

Communicated by Jimmie D. Lawson.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kumar, V., Ross, K.A. & Singh, A.I. An Addendum to “Hypergroup deformations of semigroups”. Semigroup Forum 99, 196–197 (2019). https://doi.org/10.1007/s00233-019-10023-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-019-10023-2