I'm working on learning Persistent Homology from "Computational Topology: An Introduction" by Herbert Edelsbrunner and John L. Harer. In section VII.1, Persistant Homology, they start with a monotonic function $f:K\to \mathbb R$ (monotonic in the sense that if $\sigma _i$ is a face of $\sigma _j$, then $f(\sigma _i)\le f(\sigma _j)$).
Using this function, they then reorder the simplices of $K$, putting them all together from every level as $\sigma _1$ to $\sigma _n$ by the rule $i<j$ if $f(\sigma _i)<f(\sigma _J)$ or if $\sigma _i$ is a face of $\sigma _j$
They then define an $n$ by $n$ 0-1 matrix $\partial$ by $\partial [i,j]=1$ if $\sigma _i$ is a codomension 1 face of $\sigma _j$
This puts the rows and columns as the simplicies and the boundary of a simplex is recorded in its column. They claim then to have an algorithm to reduce $\partial$ to another 0-1 matrix $R$. They do this with a function $low(j)$ which is defined on nonzero columns $j$ as the lowest row that there exists a 1, and is undefined on 0 columns. I have two questions on this process. First, the algorithm:
$$R=\partial$$ $$\text {for } j=1 \text { to } m \text { do}$$ $$\text { while there exists }j_0 <j\text { with }low(j_0)=low(j)\text { do}$$ $$\text {add column }j_o \text { to column }j $$ $$\text {endwhile}$$ $$\text {endfor}$$ First question that popped up: How does this end up with a 0-1 matrix, if we are adding two columns with 1's in the same row? My professor suggested they probably meant addition mod 2, but I thought I'd check to see if anyone knows for certain.
Next, they say that in matrix notation, the algorithm computes the reduced matrix as $R=\partial \cdot V$. I'm not sure what $V$ is here, or how that claim is made. The only thing about it is a bit later where they say "the $j$-th column of $V$ encodes the columns in $\partial$ that add up to give the $j$-th column in $R$. If anyone can clarify that, I'd appreciate it.