Combinatorial aspects of n-ary polygroups and n-ary color schemes

In this paper, the concepts of full conjugation relations and color schemes on n-ary polygroups are introduced. An n-ary color scheme on an n-ary polygroup is a generalization of a color scheme on a polygroup. We define the extension of two n-ary polygroups as a construction for building a “big” n-ary polygroup from two “small” ones. Moreover, we introduce the class of n-ary groups by a full conjugation relation on n-ary polygroups. In addition, we prove that the direct product of two n-ary color schemes and the class of n-ary groups by a full conjugation relation on n-ary polygroups are chromatic.