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This book provides a comprehensive algebraic treatment of hypergroups, as defined by F. Marty in 1934. It starts with structural results, which are developed along the lines of the structure theory of groups. The focus then turns to a number of concrete classes of hypergroups with small parameters, and continues with a closer look at the role of involutions (modeled after the definition of group-theoretic involutions) within the theory of hypergroups. Hypergroups generated by involutions lead to the exchange condition (a genuine generalization of the group-theoretic exchange condition), and this condition defines the so-called Coxeter hypergroups. Coxeter hypergroups can be treated in a similar way to Coxeter groups. On the other hand, their regular actions are mathematically equivalent to buildings (in the sense of Jacques Tits). A similar equivalence is discussed for twin buildings. The primary audience for the monograph will be researchers working in Algebra and/or Algebraic Combinatorics, in particular on association schemes.
List of contents
1 Basic Facts.- 2 Closed Subsets.- 3 Elementary Structure Theory.- 4 Subnormality and Thin Residues.- 5 Tight Hypergroups.- 6 Involutions.- 7 Hypergroups with a Small Number of Elements.- 8 Constrained Sets of Involutions.- 9 Coxeter Sets of Involutions.- 10 Regular Actions of (Twin) Coxeter Hypergroups.
About the author
Paul-Hermann Zieschang received his doctoral degree from the Christian-Albrechts-Universität zu Kiel (Germany), where he also completed his Habilitation. After holding temporary positions at Kansas State University and Kyushu University (Fukuoka), he joined the Department of Mathematics of the University of Texas at Brownsville. Since 2015, he has been Full Professor at the University of Texas Rio Grande Valley. The focus of his mathematical research is on finite groups, association schemes, and hypergroups.
Report
This well-written book focuses on the abstract algebraic notion of hypergroup. ... This book is a very interesting, readable and descriptive text in the algebraic theory of hyperstructures and can be a good reference for researchers working in algebra. (Dariush Heidari, zbMATH 1552.20002, 2025)
The book is written clearly, with attention to detail in both definitions and proofs, and the editorial choices make this book reader-friendly. (Jan Galuszka, Mathematical Reviews, December, 2024)
