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.