Products of multidimensional matrices, stochastic matrices, and permanents

The paper is devoted to four basic multidimensional matrix operations (outer product, Kronecker product, contraction, and projection) and two derivative operations (dot and circle products). It is studied the interrelations between these operations, some of their algebraic properties, and their action on $k$-stochastic matrices. Also, it is proved several relations on the permanents of products of multidimensional matrices. In particular, it is shown that the permanent of the dot product of nonnegative multidimensional matrices is not less than the product of their permanents. Another result of the paper is that inequalities on the Kronecker product of nonnegative $2$-dimensional matrices cannot be extended to the multidimensional case.