APPLICATION OF FUNDAMENTAL RELATIONS ON n-ARY POLYGROUPS

The class of n-ary polygroups is a certain subclass of n-ary hypergroups, a generalization of Dornte n-ary groups and a generalization of polygroups. The β*-relation and the γ*-relation are the smallest equivalence relations on an n-ary polygroup P such that P/β* and P/γ* are an n-ary group and a commutative n-ary group, respectively. We use the β*-relation and the γ*-relation on a given n-ary polygroup and obtain some new results and some fundamental theorems in this respect. In particular, we prove that the relation is transitive on an n-ary polygroup.