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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.


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

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  2. Kumar, V., Ross, K.A., Singh, A.I.: Hypergroup deformations of semigroups. Semigroup Forum (2019).

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

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Correspondence to Vishvesh Kumar.

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Communicated by Jimmie D. Lawson.

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Kumar, V., Ross, K.A. & Singh, A.I. An Addendum to “Hypergroup deformations of semigroups”. Semigroup Forum 99, 196–197 (2019).

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