**5. Conclusions**

For physical applications, for instance, the particle content of any elementary particle model is connected with the direct decomposition of its gauge symmetry group. Thus, the proposed generalization of the direct product can lead to principally new physical models having unusual mathematical properties.

For mathematical applications, further analysis of the direct product constructions introduced here and their examples for polyadic rings and fields would be interesting, and could lead to new kinds of categories.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Acknowledgments:** The author is deeply grateful to Vladimir Akulov, Mike Hewitt, Vladimir Tkach, Raimund Vogl and Wend Werner for numerous fruitful discussions, important help and valuable support.

**Conflicts of Interest:** The authors declare no conflict of interest.
