Geodesic
Completion of partially ordered sets
Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 1, pp. 59-67

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The paper considers a generalization of the standard completion of a partially ordered set through the collection of all its lower sets.
Keywords: quantale module, reflective category, Q-poset 
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Solovyov, Sergey. Completion of partially ordered sets. Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 1, pp. 59-67. http://geodesic.mathdoc.fr/item/DMGAA_2007_27_1_a3/