Properties of strong dependance $n$-ary semigroups

The paper presents results about the structure of strongly dependent $n$-ary operations on a finite set that satisfy the associativity identities for the $n$-ary semigroup, $n\ge 3$. It is shown that their description is reduced to the description of binary semigroups with a unit satisfying certain properties. The information is based on the proof of analogues of the Post and Gluskin–Hossu theorems for the case of strongly dependent operations. It is also proved that any strong dependence binary semigroup is a monoid. A description of autotopy groups of strongly dependent $n$-ary semigroup is also given.