Application of fundamental relations on n-ary polygroups

Document Type : Research Paper


1 Payame Noor University

2 Yazd University


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