Hypergroups
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Beschreibung
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.
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.
The text provides a direct path from elementary algebraic and combinatorial observations to research problems The book is the first attempt to systematically develop a structure theory of hypergroups with a neutral element The approach relates to different algebraic and geometric objects (groups, association schemes, buildings)
Autor*in
Paul-Hermann Zieschang
Themen in »Hypergroups«
20N20, 20A05, 05E30 51E24, 51E05, 51E15, 51F15, 20E42, 20F55, 20P99, 05Exx hypergroup association scheme group building projective space combinatorics
Stimmen zu »Hypergroups«
“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 Gałuszka, Mathematical Reviews, December, 2024)
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