# Generalized Inverses of Boolean Tensors via Einstein Product

## TL;DR

This paper extends the theory of boolean matrices to boolean tensors using the Einstein product, introducing generalized inverses, characterizations, and tensor rank definitions to enhance multiway data analysis.

## Contribution

It develops a new framework for generalized inverses of boolean tensors with the Einstein product, including characterizations, equivalence results, and a tensor rank concept.

## Key findings

- Characterizations of generalized inverses for boolean tensors
- Equivalence results on boolean tensors
- Definition of tensor rank via space decomposition

## Abstract

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.