Cyclic $n$-ary groups and their generalizations

The authors define and study $m$-semicyclic $n$-ary groups for any divisor $m-1$ of natural number $n-1$. The class of all $m$-semicyclic $n$-ary groups is included in the class of all $m$-semiabelian $n$-ary groups identified by E. Post. Moreover, the class of all $m$-semicyclic $n$-ary groups includes the class of all cyclic $n$-ary groups and belongs to the class of all semicyclic $n$-ary groups. New criteria of cyclicity for $n$-ary group and for $m$-semicyclicity of $n$-ary group formulated by one of the subgroups of the universal covering group of Post are determined.