Weakly invertible $n$-quasigroups

We study the $n$-quasigroups $(n \geqslant3)$ with the following property weak invertibility. If on any two sets of $n$ arguments with the equal initials, equal ends, but with different middle parts (of the same length), the result of the operation is the same, then for any identical beginnings (of a other length), with the previous middle parts and for any identical ends (the corresponding length), the result of the operation will be the same. For such $n$-quasigroups An analog of the Post-Gluskin-Hoss theorem is proved, which reduces the operation of an $n$-quasigroup to a group one. The representation of the $n$-quasigroup operation proved by the theorem with the help of the automorphism of the group turned out to occur in weaker (and quite natural) assumptions, rather than the associativity and $(i, j)$-associativity required earlier. Well-known $(i, j)$-associative $n$-quasigroups satisfy the condition of weak invertibility.