Idempotents in polyadic groupoids of special form

The article focuses on idempotents in polyadic groups of a special form. The main result was obtained for $l$-ary group of a special form, i. e. for polyadic group with $l$-ary operation $\eta_{s,\sigma,k}$, that is called polyadic operation of a special form and is defined on Cartesian power $A^k$ of $n$-ary group $\langle A,\eta\rangle$ by substitution $\sigma\in\mathbf{S}_k$, satisfying the condition $\sigma^1=\sigma$, and $n$-ary operation $\eta$. As corollaries there were obtained the results for polyadic groups of a special form with $(2s+1)$-ary operation $\eta_{s,\sigma,k}$, which is defined on Cartesian power $A^k$ of ternary group $\langle A,\eta\rangle$ by substitution $\sigma\in\mathbf{S}_k$ which satisfies the condition $\sigma^{2s+1}=\sigma$, and ternary operation $\eta$.