Suppose we have two directed graphs, A and B, each represented as a set of ordered pairs (x,y) of natural numbers, such that x and y represent vertices and (x,y) represents a directed edge from x to y.
Let's define A ⊆ B to be true iff there exists an injective function F: A → B such that if two pairs in A share the same first, resp. second element, then their images also share the same first, resp. second element. That is, one could take the graph A and overlay it atop B and it would match some part of B.
A variant of this subset relation might be more strict in requiring that the images of A's vertices under F do not have any edges between them that are not images of edges in A. By the former relation, a graph of three vertices in the shape a->b->c would be a subset of a triangle with all edges bidirectional - but not by the latter.
Either way, one could construct a lattice of graphs this way, with each point in the lattice corresponding to a graph, and given L(X) takes graph X to its corresponding lattice point, then A ⊆ B would imply that L(A) ≤ L(B). Meet and join could then be defined as well, though I do not know if they are guaranteed to be unique - I suspect not.
So, my question is basically, has this been done? Does anyone study lattices of graphs ordered under one of these types of inclusion, or some other I haven't thought of? Is there even anything interesting TO study there? If so, I'd love to see some references to papers or books about this subject.