In this paper we primarily study monomial ideals and their minimal free
resolutions by studying their associated LCM lattices. In particular, we
formally define the notion of coordinatizing a finite atomic lattice P to
produce a monomial ideal whose LCM lattice is P, and we give a complete
characterization of all such coordinatizations. We prove that all relations in
the lattice L(n) of all finite atomic lattices with n ordered atoms can be
realized as deformations of exponents of monomial ideals. We also give
structural results for L(n). Moreover, we prove that the cellular structure of
a minimal free resolution of a monomial ideal M can be extended to minimal
resolutions of certain monomial ideals whose LCM lattices are greater than that
of M in L(n).