Various algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary ("polyadization" and partial arity freedom principle), and the relations between operations, arising from the structure definitions, lead to restrictions of their arity shapes ("quantization"). This unified procedure is applied to one-set and two-set algebraic structures, as well as to Hopf algebras and tensor categories, which gives new unusual properties and enriched objects absent in the ordinary binary structures.