Atomic, upper semi-modular lattices, or geometric lattices, have been well studied because of their connections with matroids, projective geometries, subspaces of vector spaces, and other mathematical areas of interest. Closure lattices, although also very regular, are neither atomic nor semi-modular and so have been largely ignored. Yet, they arise naturally in the study of partial order relations, or acyclic directed graphs.
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Pfaltz, John. "Closure Lattices." University of Virginia Dept. of Computer Science Tech Report (1994).
University of Virginia, Department of Computer Science