In this paper a new form of the Hosszú--Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszú--Gluskin chain formula is not unique and can be generalized (``deformed'') using a parameter q which takes special integer values. A version of the ``q-deformed'' analog of the Hosszú--Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The ``q-deformed'' homomorphism theorem is also given.