Arity shape of polyadic algebraic structures

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to the restrictions which determine their possible arity shapes and lead us to formulate a partial arity freedom principle. Polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered.
Elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic $C^{*}$-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators.
It is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 7.17), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander–Parkin–Selfridge conjecture and Fermat's Last Theorem are formulated. For polyadic numbers neither of these statements holds. Polyadic versions of Frolov's theorem and the Tarry–Escott problem are presented.