Combinatorial aspects of -ary polygroups and -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.