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 D{"o}rnte $n$-ary groups and a generalization of polygroups. The$beta^*$-relation and the $gamma^*$-relation are the smallest equivalence relations on an $n$-ary polygroup $P$ such that $P/beta^*$ and $P/gamma^*$ are an $n$-ary group and a commutative $n$-ary group, respectively. We use the $beta^*$-relation and the $gamma^*$-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 $gamma$ is transitive on an $n$-ary polygroup.