Minimal monomial ideals and linear resolutions

A minimal monomial ideal is the combinatorially simplest monomial ideal whose lcm-lattice equals a given finite atomic lattice $\widehat{L}$. The minimal ideal inherits many nice properties of any ideal $I$ whose lcm-lattice also equals $\widehat{L}$, e.g. Cohen-Macaulayness and the dual property of having a linear resolution. Conversely, any ideal having a linear resolution is shown to be (essentially) minimal.