EDIT: I'd like to say this algebra is more general since in the lattice sense, any undirected graph is just equivalent to a poset in which all of the elements are equal (e.g. since x->y+y->x ≅ {x}, where x=y), but there's obviously more interesting structure than that within the graph. In general, there's only truly interesting structure within the graph when the graph has a topological sort (anything else gives elements which are equal under the partial order).
There may be some weirdness where we can take subsets of vertices/edges of graphs and then endow those subsets with lattice structure, which is somehow isomorphic to the original graph, but this is again, not obvious to me how it would be done.