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.
