Periodic semi-groups in the ring of residue classes

In this work the structure periodic semi-groups $S(m;n)$ have been learnt which are given by definite coorelation $X^n=X$, $n>1$ in the ring $Z_m$ of residue classes the modulo $m$. The main result which determines the structure $S(m;n)$ is expressed by correlation: $S(m;n)=\cup_{i\in I(m)}G(i)$, $G(i_1)\cap G(i_2)=\varnothing$, $i_1,i_2\in I_m$, $i_1\neq i_2$, where $G(i)$ — maximal undergroup (in sence [6]) is generated by idempotent $i$ of the semi-lattice $I(m)\subset Z_m$.