The authors consider a problem of uniqueness of a tensor decomposition, related to tensors \(\mathcal X\) of type \(I\times J\times K\). In particular, they are interested in decompositions which arise by unfolding \(\mathcal X\) to a matrix \(X\) and considering representations of type \(X=(A\odot B)S^t\), where \( A\), \(B\), \(S\) are matrices of size \(I\times R\), \(J\times R\), \(K\times R\), respectively, and \(\odot\) represents the column-wise Kronecker product. The authors consider the uniqueness of the decomposition when the entries of the matrix \(A\) are subject to monomial equality constraints, i.e., the entries must satisfy equations of type \(a_{p_1r}\cdots a_{p_Lr} = a_{s_1r}\cdots a_{s_Lr}\), and similar constraints hold on the entries of \(S\). Such monomial equality constraints arise naturally in decomposition problems related to signal processing or binary latent variable modeling. The authors extend existing methods for detecting the uniqueness of a general rank decomposition to tensor decompositions with monomial equality constraints. The block term decomposition method allows to restrict the monomial constraints to rank constraint. Thus, the authors are able to find several conditions that guarantee the uniqueness of the decomposition. The authors observe that the same methods can provide uniqueness results even for weighted decompositions of type \(X=(D*A\odot B)S^t\), where \(D\) is a matrix of weights, and \(*\) denotes the entry-wise (Hadamard) product.