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Completion of partially ordered sets. (English) Zbl 1141.18005

A quantale \(Q\) is a complete lattice with an associative binary operation \((\cdot )\) such that \(a\cdot -\) and \(-\cdot a\) both preserve arbitrary joins for all \(a\in Q\). It is well known that the category \({\mathcal {JCP}os}\) of complete lattices and join-preserving maps is reflective in the category \({\mathcal P}os\) of posets and order-preserving maps. In this paper the author generalizes this result. More precisely, he proves that if \(Q\) is a quantale, then the category \(Q\)-\({\mathcal M}od\) of modules over \(Q\) is reflective in \({\mathcal P}os\). Also it is shown that for a quantale \(Q\), the category \(Q\)-\({\mathcal M}od\) is reflective in the category \(Q\)-\({\mathcal P}os\) of \(Q\)-posets.

MSC:

18B35 Preorders, orders, domains and lattices (viewed as categories)
06B23 Complete lattices, completions
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