Polyadic Algebraic Structures

The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting with works by Cayley, Sylvester, Kasner, Lehmer, Post, etc. Their idea was to take a single set, closed under one binary operation having special properties, the so called structure, and then to "generalize" it by increasing the arity of the operation which was called a polyadic operation and the corresponding algebraic structure polyadic as well. However, until now, a general approach to polyadic concrete many-set algebraic structures was absent. Here we propose to investigate algebraic structures in the "concrete way" and provide the consequent "polyadization" of each operation, preserving "interactions " between them, starting from group-like, module-like and algebra-like structures and finishing with the coalgebraic and Hopf algebra structures. Polyadic analogs of homomorphisms which change the arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide a new polyadic approach to quantum groups, polyadic generalization of the Yang-Baxter equation and its constant solutions, introduce higher braidings and medialings, as well as polyadic tensor and braided categories.

Suitable for university students of advanced level algebra courses and mathematical physics courses.