Direct product of $n$-ary groups

The notion of $n$-ary group is a generalization of the binary group so many of the results from the theory of groups have $n$-ary analogue in theory of $n$-ary groups. But there are significant differences in these theories. For example, multiplier of the direct product of $n$-ary groups does not always have isomorphic copy in this product (in paper there is an example). It is proved that the direct product $\prod_{i\in I}\langle A_i,f_i\rangle$ $n$-ary groups has $n$-ary subgroup isomorphic to $\langle A_j,f_j\rangle$ ($j\in I$), then and only when there is a homomorphism of $\langle A_j,f_j\rangle$ in $\prod_{i\in I,i\ne j}\langle A_i,f_i\rangle.$ Were found necessary and sufficient conditions for in direct product of $n$-ary groups, each of the direct factors had isomorphic copy in this product and the intersection of these copies singleton (as well as in groups) — each direct factor has a idempotent.
For every $n$-ary group, can define a binary group which helps to study the $n$-ary group, that is true Gluskin–Hossu theorem: for every $n$-ary group of $\langle G,f\rangle$ for an element $e\in G$ can define a binary group $\langle G,\cdot\rangle$, in which there will be an automorphism $\varphi(x)=f(e,x,c_1^{n-2})$ and an element $d=f(\overset{(n)}{e})$ such that the following conditions are satisfied:
\begin{align*} &f(x_1^n)=x_1\cdot\varphi(x_2)\cdot\ldots\cdot\varphi^{n-1}(x_n)\cdot d, ~~ x_1,x_2,\ldots,x_n\in G;\qquad\qquad\qquad\!(4)\\ &\varphi(d)=d;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad\,\,\,(5)\\ &\varphi^{n-1}(x)=d\cdot x\cdot d^{-1}, ~~ x\in G.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(6) \end{align*}
Group $\langle G,\cdot\rangle$, which occurs in Gluskin–Hossu theorem called retract $n$-ary groups $\langle G,f\rangle$.
Converse Gluskin–Hossu theorem is also true: in any group $\langle G,\cdot \rangle $ for selected automorphism $\varphi $ and element $d$ with the terms (5) and (6), given $n$-ary group $\langle G,f \rangle $, where $ f $ defined by the rule (4). A $ n $-ary group called ($\varphi, d $)-defined on group $\langle G, \cdot \rangle $ and denote $der_{\varphi, d} \langle G, \cdot \rangle $.
Was found connections between $n$-ary group, ($\varphi, d$)-derived from the direct product of groups and $n$-ary groups that ($\varphi_i, d_i $)-derived on multipliers of this product: let $\prod_{i \in I} \langle A_i, \cdot_i \rangle$ — direct product groups and $\varphi_i$, $d_i$ — automorphism and an element in group $\langle A_i, \cdot_i \rangle$ with the terms of (5) and (6) for any $i \in I$. Then
$$der_{\varphi, d} \prod_{i \in I} \langle A_i, \cdot_i \rangle = \prod_{i \in I} der_{\varphi_i, d_i} \langle A_i, \cdot_i \rangle,$$
where $\varphi$ – automorphism of direct product of groups $\prod_{i \ in I} \langle A_i, \cdot_i \rangle $, componentwise given by the rule: for every $a \in \prod_{i \in I} A_i $, $ \varphi (a) (i) = \varphi_i (a (i)) $ (called diagonal automorphism), and $d (i) = d_i $ for any $i \in I$.
In the theory of $n$-ary groups indecomposable $n$-ary groups are finite primary and infinite semicyclic $n$-ary groups (built by Gluskin–Hossu theorem on cyclic groups). We observe $n$-ary analogue indecomposability cyclic groups. However, unlike groups, finitely generated semi-abelian $n$-ary group is not always decomposable into a direct product of a finite number of indecomposable semicyclic $n$-ary groups. It is proved that any finitely generated semi-abelian $n$-ary group is isomorphic to the direct product finite number of indecomposable semicyclic $n$-ary groups (infinite or finite primary) if and only if in retract this $n$-ary group automorphism $\varphi$ from Gluskin–Hossu theorem conjugate to some diagonal automorphism.
Bibliography: 18 titles.