Polygroup Theory and Related Systems

This monograph is devoted to the study of Polygroup Theory. It begins with some basic results concerning group theory and algebraic hyperstructures, which represent the most general algebraic context, in which reality can be modeled. Most results on polygroups are collected in this book. Moreover, this monograph is the first book on this theory. The volume is highly recommended to theoreticians in pure and applied mathematics.

Contents:

• A Brief Excursion into Group Theory:

  • Introduction
  • The Abstract Definition of a Group and Some Examples
  • Subgroups
  • Normal Subgroups and Quotient Groups
  • Group Homomorphisms
  • Permutation Groups
  • Direct Product
  • Solvable and Nilpotent Groups

• Hypergroups:

  • Introduction and Historical Development of Hypergroups
  • Definitions and Examples of Hypergroups
  • Some Kinds of Subhypergroups
  • Homomorphisms of Hypergroups
  • Regular and Strongly Regular Relations
  • Complete Hypergroups
  • Join Spaces

• Polygroups:

  • Definition and Examples of Polygroups
  • Extension of Polygroups by Polygroups
  • Subpolygroups and Quotient Polygroups
  • Isomorphism Theorems of Polygroups
  • γ* Relation on Polygroups
  • Generalized Permutations
  • Permutation Polygroups
  • Representation of Polygroups
  • Polygroup Hyperrings
  • Solvable Polygroups
  • Nilpotent Polygroups

• Weak Polygroups:

  • Weak Hyperstructures
  • Weak Polygroups as a Generalization of Polygroups
  • Fundamental Relations on Weak Polygroups
  • Small Weak Polygroups

• Combinatorial Aspects of Polygroups:

  • Chromatic Polygroups
  • Polygroups Derived from Cogroups
  • Conjugation Lattice

Readership: Graduate students and researchers in algebraic hyperstructures and applications.