In some project, I am comparing some sets on the basis of the elements they contain. My data can be represented as a binary matrix $M$ with sets in rows and elements in colunms. Entry $M_{ij} = 1$ if set $i$ contains elements $j$ and $0$ otherwise.
I am wondering if there exists some theory about switching the roles of sets and elements here, which would correspond to transposing the matrix $M$. This kind of approach would be a "dual view" of the data. Instead of describing sets with the list of elements they contain, we would describe elements with the list of sets they are contained in.
Does that ring a bell to someone?