  We use the lcm-lattice of a monomial ideal to study its minimal free
resolutions. A new concept called a Taylor basis of a minimal free resolution
is introduced and then used throughout the paper. We give a method of
constructing minimal free resolutions of a monomial ideal from its lcm-lattice,
which is called the atomic lattice resolution theory. Some applications of this
theory is given. As the main application, we rewrite the theory of poset
resolutions, and we obtain an approximation formula for minimal free
resolutions of all monomial ideals.
