Strange use of disjunction symbol in matrix equation

Question

I am referring to this source: 3.2 Distance matrix D

I don't understand what is meant with the following matrix equation, especially the use of the disjunction symbol:

Could you please provide an explanation and, if possible, give some kind of step-by-step recipe what to do with the adjacency matrix $A$.

Answer 1

The link goes to a page that is not viewable, but I suspect that for nonnegative integer matrices $B$ and $C$, $B \vee C$ means to replace each positive entry of the matrix sum with $1$. Explicitly, $$(B \vee C)_{r,c}=B_{r,c} \vee C_{r,c}=\min(B_{r,c} + C_{r,c}, 1).$$ So $$\left(\bigvee_{j=0}^i A^j\right)_{r,c}=\min\left(\sum_{j=0}^i (A^j)_{r,c}, 1\right).$$

Note that the $(r,c)$ entry of $A^j$ records the number of walks of length $j$ from node $r$ to node $c$.