Geodesic
On interval decomposition lattices
Discussiones Mathematicae. General Algebra and Applications, Tome 24 (2004) no. 1, pp. 95-114

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Intervals in binary or n-ary relations or other discrete structures generalize the concept of interval in an ordered set. They are defined abstractly as closed sets of a closure system on a set V, satisfying certain axioms. Decompositions are partitions of V whose blocks are intervals, and they form an algebraic semimodular lattice. Lattice-theoretical properties of decompositions are explored, and connections with particular types of intervals are established.
Keywords: interval, closure system, modular decomposition, semimodular lattice, partition lattice, strong set, lexicographic sum 
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Foldes, Stephan; Radeleczki, Sándor. On interval decomposition lattices. Discussiones Mathematicae. General Algebra and Applications, Tome 24 (2004) no. 1, pp. 95-114. http://geodesic.mathdoc.fr/item/DMGAA_2004_24_1_a6/