  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 restrictions which determine their possible arity shapes and
lead us to the partial arity freedom principle. In this manner, polyadic vector
spaces and algebras, dual vector spaces, direct sums, tensor products and inner
pairing spaces are reconsidered. As one application, 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. Another application is connected with number theory, and it is shown
that congruence classes are polyadic rings of a special kind. Polyadic numbers
are introduced (see Definition 6.16), 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 nonderived polyadic ring operations (on polyadic numbers) neither of these
statements holds, and counterexamples are given. Finally, a procedure for
obtaining new solutions to the equal sums of like powers equation over polyadic
rings by applying Frolov's theorem to the Tarry-Escott problem is presented.
