  We present a construction of a family of independent sets for a finite,
atomic and graded lattice generalizing the known cryptomorphism between
geometric lattices and matroids. This family then gives rise to an embedding
theorem into geometric lattices preserving the set of atoms. Lastly we apply
these theorems to the concept of adjoint matroids obtaining a characterization
and proving a conjecture on the combinatorial derived madtroid for uniform and
co-rank 3 matroids.
