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The paper contains an extension of Malyshev theorem for  n -ary quasigroups with a right or left weak invertibility property to the case of strongly dependent  n -ary operations. As a corollary a new proof of Gluskin–Hosszú theorem for strongly dependent  n -ary semigroups is obtained.

Analogues of Gluskin–Hosszú and Malyshev theorems for strongly dependent n -ary operations