I want to enumerate the monomorphisms of finite-dimensional $\mathbb{F}_2$-algebras.
Of course, each such monomorphism is a linear map between finite-dimensional $\mathbb{F}_2$-vector spaces, and so can be represented as a matrix with entries in $\mathbb{F}_2$. Since we are looking just at mononorphisms, the number of rows must be greater than or equal to the number of columns, and the columns must be linearly independent over $\mathbb{F}_2$.
But how can I identify those matrices corresponding to algebra homomorphisms? Is there a simple way to do this?
I hope to enumerate based on the dimension of each algebra. If $A$ and $B$ have dimension $m$ and $n$ respectively, then when $(m,n)$ is $(0,0),(0,1),(1,1),(0,2)$, there is clearly only one such monomorphism. But what about $(1,2)$?
There can be at most 3 monomorphisms, with entries $(1,0)$, $(0,1)$, $(1,1)$ (these should be written as $2\times 1$ matrices). But where can I go from here?