Imagine a collection of separate objects and several sets.
These sets can be represented using a logical matrix.
$M = \begin{bmatrix} 1 & 1 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix}$
Each line corresponds to a set, and each column corresponds to an object.
So $M_{i,j} = 1$ if $Set_i$ contains $Object_j$, else $0$.
I wondered if this kind of matrix, used to represent sets, had a special name? And if, in addition, there was some interesting properties?
For example, to represent a graph, one can use an adjacency matrix, which is a logical matrix too. By raising it to the n-th power, each $x_{i,j}$ obtained indicates the number of existing n-length paths from i to j.
I am in search of similar property (which may be about subset or superset for example), or a name to this matrix that would allow me to further my research.
In computer science, this representation of subsets is known under various names: bit array, bit string, bitmap, bitset or bit vector. It is extremely compact and Boolean operations can be implemented in a very efficient way, but it is limited to small sets.
From this point of view, your matrices might be called arrays of bitsets.