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The lattice of completions of an ordered set. (English) Zbl 0888.06003

The join dense completions of an ordered set \(P\) form a complete lattice \(K(P)\). Its least element is \(\mathcal O(P)\), the lattice of all order ideals of \(P\), and its greatest element is \(\mathcal M(P)\), the Dedekind-MacNeille completion of \(P\). It is isomorphic to an ideal of the lattice of all closure operators on \(\mathcal O(P)\). Some local structural properties of lattices of closure operators on complete lattices are proved. These are inherited by \(K(P)\) as well. In particular, if \(K(P)\) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.
Reviewer: J.Niederle (Brno)

MSC:

06B23 Complete lattices, completions
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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