Zero-One Matrix (0-1 Matrix)

A 0-1 Matrix is a matrix whose elements are either 0 or 1.

Let $A=(a_{ij}{)}_{m\times n},B=(b_{ij}{)}_{m\times n}$ be 0-1 matrices, we define

$$A\vee B=(a_{ij}\vee b_{ij}{)}_{m\times n},A\wedge B=(a_{ij}\wedge b_{ij}{)}_{m\times n}$$

and

$$A\odot B=(c_{ij}{)}_{m\times n},\text{where}{c}_{ij}=(a_{i1}\wedge b_{1j})\vee (a_{i2}\wedge b_{2j})\vee \cdots \vee (a_{ik}\wedge b_{kj})$$

which is called the Boolean Product of $A$ and $B$

Tip

We can see Boolean Product as a normal matrix multiplication with $+$ replaced by $\vee$ and $\times$ replaced by $\wedge$

Based on this, we can also define the $r$-th Boolean power of $A$ as

$$A^{[r]}=\underset{A\text{ occurs }r\text{ times}}{\underbrace{A\odot A\odot \cdots \odot A}}$$