$n$-ary polygroups are 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 $gamma^*$-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. In this paper, 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 the relation $gamma$ is transitive on an $n$-ary polygroup.
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