Minimal monomial ideals and linear resolutions

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A minimal monomial ideal is the combinatorially simplest monomial ideal whose lcm-lattice equals a given finite atomic lattice L. The minimal ideal inherits many nice properties of any ideal I whose lcm-lattice also equals 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.