Generalized Inverses of Boolean Tensors via Einstein Product
For researchers in discrete mathematics and data analysis, this work offers a theoretical foundation for handling multidimensional Boolean data, but it is purely theoretical with no experimental validation.
This paper extends the theory of generalized inverses from Boolean matrices to Boolean tensors using the Einstein product, providing characterizations and equivalence results, and defining tensor rank via space decomposition.
Applications of the theory and computations of boolean matrices are of fundamental importance to study a variety of discrete structural models. But the increasing ability of data collection systems to store huge volumes of multidimensional data, the boolean matrix representation of data analysis is not enough to represent all the information content of the multiway data in different fields. From this perspective, it is appropriate to develop an infrastructure that supports reasoning about the theory and computations. In this paper, we discuss the generalized inverses of the Boolean tensors with the Einstein product. Further, we elaborate on this theory by producing a few characterizations of different generalized inverses and several equivalence results on boolean tensors. In addition to these, we define the rank of a boolean tensor through space decomposition.