Hypergroups / Paul-Hermann Zieschang.
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,...
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| Language: | English |
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| 245 | 1 | 0 | |a Hypergroups / |c Paul-Hermann Zieschang. |
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| 505 | 0 | |a Intro -- Preface -- Contents -- 1 Basic Facts -- 1.1 Neutral Elements and Inversion Functions -- 1.2 Products -- 1.3 Complex Products -- 1.4 Thin Elements -- 1.5 Groups and Thin Hypergroups -- 1.6 Actions of Hypergroups -- 1.7 Hypergroups Admitting Regular Actions -- 1.8 Association Schemes -- 2 Closed Subsets -- 2.1 Basic Facts -- 2.2 Dedekind Modularity -- 2.3 Generating Sets -- 2.4 Commutators -- 2.5 Conjugation -- 2.6 The Thin Radical -- 2.7 Foldings -- 3 Elementary Structure Theory -- 3.1 Centralizers and Normalizers -- 3.2 Sufficient Conditions for Normality -- 3.3 Strong Normality | |
| 505 | 8 | |a 3.4 Quotients -- 3.5 Computations in Quotients -- 3.6 Hypergroup Homomorphisms -- 3.7 The Isomorphism Theorems -- 3.8 Wreath Products -- 3.9 Hypergroup Isomorphisms and Regular Actions -- 4 Subnormality and Thin Residues -- 4.1 Subnormal Chains -- 4.2 Subnormal Series -- 4.3 Composition Series -- 4.4 The Thin Residue -- 4.5 Thin Residues of Thin Residues -- 4.6 Residually Thin Hypergroups -- 4.7 Finite Residually Thin Hypergroups -- 4.8 Solvable Hypergroups -- 5 Tight Hypergroups -- 5.1 Tight Hypergroup Elements -- 5.2 The Set S -- 5.3 The Sets a*b ∩ Fc and Sa,b(Fc) | |
| 505 | 8 | |a 5.4 The Sets b f1b* ∩ Fa and Sb.(f1,...,fn)(Fa) -- 5.5 Structure Constants of Finite Tight Hypergroups -- 5.6 Rings Arising from Finite Tight Hypergroups -- 5.7 Some Arithmetic in Finite Metathin Hypergroups -- 5.8 Finite Metathin Hypergroups and Prime Numbers -- 6 Involutions -- 6.1 Basic Facts -- 6.2 Closed Subsets Generated by an Involution, I -- 6.3 Closed Subsets Generated by an Involution, II -- 6.4 Closed Subsets Generated by an Involution, III -- 6.5 Length Functions Defined by Sets of Involutions -- 6.6 Hypergroups Generated by Two Involutions -- 6.7 Dichotomy and the Exchange Condition | |
| 505 | 8 | |a 6.8 Projective Hypergroups -- 7 Hypergroups with a Small Number of Elements -- 7.1 Hypergroups of Cardinality at Most 3 -- 7.2 Non-Symmetric Hypergroups of Cardinality 4 -- 7.3 Some Hypergroups of Cardinality 6, I -- 7.4 Some Hypergroups of Cardinality 6, II -- 7.5 Non-Normal Closed Subsets Missing Four Elements -- 7.6 A Characterization of H6,1 -- 8 Constrained Sets of Involutions -- 8.1 Basic Results -- 8.2 Constrained Sets of Involutions and Cosets -- 8.3 Constrained Sets of Involutions and Thin Elements -- 8.4 Constrained Sets of Involutions and Dichotomy, I | |
| 505 | 8 | |a 8.5 Constrained Sets of Involutions and Dichotomy, II -- 8.6 Constrained Sets of Involutions and Foldings, I -- 8.7 Constrained Sets of Involutions and Foldings, II -- 9 Coxeter Sets of Involutions -- 9.1 General Observations -- 9.2 The Sets V1(U) for Subsets U of Coxeter Sets V -- 9.3 The Sets V−1(U) for Subsets U of Coxeter Sets V -- 9.4 Sets of Subsets of Coxeter Sets of Involutions -- 9.5 Spherical Coxeter Sets of Involutions -- 9.6 Subsets of Spherical Coxeter Sets of Involutions -- 9.7 Coxeter Sets of Involutions and Foldings -- 9.8 Coxeter Sets and Their Coxeter Numbers | |
| 500 | |a 9.9 Coxeter Sets and Type Preserving Bijections | ||
| 520 | |a 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. | ||
| 504 | |a Includes bibliographical references and index. | ||
| 588 | 0 | |a Online resource; title from PDF title page (SpringerLink, viewed November 16, 2023). | |
| 650 | 0 | |a Hypergroups. | |
| 776 | 0 | 8 | |i Print version: |a Zieschang, Paul-Hermann |t Hypergroups |d Cham : Springer International Publishing AG,c2023 |z 9783031394881 |
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