Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X38120120401Application of fundamental relations on n-ary polygroups169184398ENS.MirvakiliPayame Noor UniversityB.DavvazYazd UniversityJournal Article20090430The class of $n$-ary polygroups is a certain subclass of $n$-ary hypergroups, a generalization of D{\"o}rnte $n$-ary<br />groups and a generalization of polygroups. The<br />$\beta^*$-relation and the $\gamma^*$-relation are the smallest<br />equivalence relations on an $n$-ary polygroup $P$ such that<br />$P/\beta^*$ and $P/\gamma^*$ are an $n$-ary group and a<br />commutative $n$-ary group, respectively.<br /> We use the $\beta^*$-relation and the $\gamma^*$-relation on a given<br />$n$-ary polygroup and obtain some new results and some<br />fundamental theorems in this respect. In particular, we prove that the relation $\gamma$ is transitive on an $n$-ary<br />polygroup.http://bims.iranjournals.ir/article_398_343a6bc9bb99db6c78480f38fbc278d8.pdf