This book provides a general, unified approach to the theory of polyadic groups, their normal subgroups and matrix representations.
The author focuses on those properties of polyadic groups which are not present in the binary case. These properties indicate a strong relationship between polyadic groups and various group-like algebras, as well as ternary Hopf algebras and n-Lie algebras that are widely used in theoretical physics.
The relationships of polyadic groups with special types of binary groups, called covering groups and binary retracts, are described. These relationships allow the study of polyadic groups using these binary groups and their automorphisms.
The book also describes the affine geometry induced by polyadic groups and fuzzy subsets defined on polyadic groups. Finally, we discuss the categories of polyadic groups and the relationships between the different varieties of polyadic groups. In many cases, we give elegant new proofs of known theorems. We also give many interesting examples and applications.
The book contains many little-known results from articles previously published in hard-to-reach Russian, Ukrainian and Macedonian journals. These articles are not in English.
Preface
Introduction
1. Basic concepts
1.1 Basic definitions and examples
1.2 Homomorphisms
1.3 Covering semigroups
1.4 Extension of homomorphisms
1.5 Post's coset theorem
1.6 Cancellable elements
1.7 Regular $n$-semigroups
1.8 Inverse $n$-semigroups
1.9 Divisibility in $n$-semigroups
1.10 Retracts
1.11 Notes on Chapter 1
2. Varieties of $n$-groups
2.1 D\"ornte's identities
2.2 Generalized D\"ornte's identities
2.3 Variety of $n$-groups
2.4 Subvariety of idempotent $n$-groups
2.5 Free $n$-groups
2.6 Notes on Chapter 2
3 Partially commutative $n$-groups
3.1 Binary quasigroups
3.2 Medial $n$-quasigroups
3.3 Medial $n$-groups
3.4 Permutable $n$-groups
3.5 $m$-semiabelian $n$-groups
3.6 Weakly semiabelian $n$-groups
3.7 Free semiabelian $n$-groups
3.8 Notes on Chapter 3
4 Subgroups
4.1 Normal $n$-subgroups
4.2 Normalizers and seminormalizers
4.3 Congruences
4.4 Nilpotent $n$-groups
4.5 Direct products of $n$-groups
4.6 Rusakov's direct product
4.7 Notes on Chapter 4
5 Cyclic $n$-groups
5.1 Orders of elements
5.2 Cyclic $n$-groups
5.3 Precyclic $n$-groups
5.4 Autodistributive $n$-groups
5.5 Distributive $n$-groups
5.6 Notes on Chapter 5
6 Automorphisms
6.1 Isotopies of $n$-groups
6.2 Autotopies of $n$-groups
6.3 Automorphisms of $n$-groups
6.4 Skew automorphisms
6.5 Splitting automorphisms
6.6 Notes on Chapter 6
7 Representations of $n$-groups
7.1 Substitutions
7.2 Bi-element representation of $n$-groups
7.3 Relations between bi-representations
7.4 Diagonal representations
7.5 Action of an $n$-group on a set
7.6 G-modules, representations and characters
7.7 Connections between representations
7.8 n-groups of small orders
7.9 Notes on Chapter 7
8 Various types of $n$-groups
8.1 $n$-grouops with the inverse property
8.2 Self-orthogonal $n$-groups
8.3 Partially ordered $n$-groups
8.4 Topological $n$-groups
8.5 Fuzzyfication of $n$-groups
8.6 Hyper $n$-groups
8.7 Notes on Chapter 8
9 Geometry induced by $n$-groups
9.1 Ternary groups and flocks
9.2 $rs$-flocks and $n$-groups
9.3 Geometry defined by flocks
9.4 Notes on Chapter 9
10 Category of $n$-groups
10.1 Covering $k$-groups of $n$-groups
10.2 The category of covering $(k+1)$-groups
10.3 $(k+1)$-ary retracts
10.4 Functors $\Psi $ and $\Phi $
10.5 Morphisms
10.6 Inductive and projective limits
10.7 Free products
10.8 Amalgamated products
10.9 Profinite $(n+1)$-groups
10.10 Notes on Chapter 10
Bibliography
Index
Biography
Wieslaw A. Dudek received his PhD in Mathematics from the Institute of Mathematics of the Moldavian Academy of Sciences, Moldova (supervisor: V.D. Belousov). A few years later, he received an academic degree, Dr Sci. (habilitation), from the Warsaw University of Technology. His research areas are universal algebra, n-ary systems (especially n-ary groups), quasigroups, algebraic logics and various types of fuzzy sets. He has published more than 150 research papers and six books cited in more than 30 monographs. His h-index is 24. He is a member of the editorial board of several mathematical journals and the main editor of the journal Quasigroups and Related Systems.
