Polyadic Groups

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.

1. Basic concepts. 2. Varieties of $n$-groups. 3. Partially commutative $n$-groups. 4. Subgroups. 5. Cyclic $n$-groups. 6. Automorphisms. 7. Representations of $n$-groups. 8. Various types of $n$-groups. 9. Geometry induced by $n$-groups. 10. Category of $n$-groups .

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.