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On pseudocomplemented and Stone ordered sets. (English) Zbl 0999.06004
The notions of distributive, pseudocomplemented and Stone ordered sets are generalisations of the analogous notions in lattice theory. In their definitions the lattice operations are replaced by upper and lower cones of subsets of ordered sets. If \(A\) is an ordered set and \(\text{DM}(A)\) is the Dedekind-MacNeille completion of \(A\), denote by \(G(A)\) the sublattice of \(\text{DM}(A)\) generated by \(A\). It is known, e.g., that \(A\) is distributive if and only if \(G(A)\) is distributive. In the paper under review it is shown that there is a distributive pseudocomplemented ordered set \(A\) such that \(G(A)\) is not pseudocomplemented. So a stronger notion of pseudocomplementedness is introduced, which is shown to be equivalent on \(A\) and on \(G(A)\).

MSC:
06A06 Partial orders, general
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