On the permanent of multidimensional matrices

The permanent of multidimensional matrices is expressed in terms of operations on elements of commutative algebra with nilpotent index 2 generators. Using a technique based on this relationship, several properties of the permanent have been proved. Various types of multidimensional permutations are considered. The permanent of multidimensional matrices is considered from the point of view of the enumeration function of multidimensional permutations.