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Hypergroups all nonidentity elements of which are involutions. (English) Zbl 1472.20150

Feldvoss, Jörg (ed.) et al., Advances in algebra. Proceedings of the southern regional algebra conference, SRAC 2017, Mobile, AL, USA, March 17–19, 2017. Cham: Springer. Springer Proc. Math. Stat. 277, 305-322 (2019).
Summary: The notion of a hypergroup generalizes the notion of a group. We introduce a hypergroup-theoretic generalization of the group-theoretic notion of an involution and characterize the hypergroups all nonidentity elements of which are involutions. Our characterization sheds new light on previous investigations in which a corresponding condition was considered within the theory of association schemes [E. R. van Dam, Des. Codes Cryptography 21, No. 1–3, 83–86 (2000; Zbl 0964.05072); the author, Theory of association schemes. Berlin: Springer (2005; Zbl 1079.05099)] and table algebras [H. I. Blau and G. Chen, Arch. Math. 105, No. 4, 313–322 (2015; Zbl 1359.16014)]. We also show in how far hypergroups all nonidentity elements of which are involutions play a role in the investigation [A. Wang, Constrained and Coxeter table algebras. DeKalb, IL: Northern Illinois University. (PhD. Thesis) (2016)] of constrained hypergroups and their relationship to Coxeter hypergroups.
For the entire collection see [Zbl 1411.13002].

MSC:

20N20 Hypergroups
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