The injective hull of ultra-quasi-metric versus \(q\)-hyperconvex hull of quasi-metric space
Given a \(T_0\)-ultra-quasi-metric space \((X,d)\), one can construct \(\nu(X,d)\), its injective hull with respect to \(T_0\)-ultra-quasi-metric spaces, see \textit{H.-P. A. Künzi} et al. [Appl. Categ. Struct. 21, No 6, 651--670 (2013; Zbl 1284.54040)], and \(\varepsilon(X,d)\), its injective hull with respect to \(T_0\)-quasi-metric spaces, see \textit{E. Kemajou} et al. [Topology Appl. 159, No. 9, 2463--2475 (2012; Zbl 1245.54023)].NEWLINENEWLINE In this paper it is observed that, if \((X,\leq)\) is a partially ordered set and \(d\) is the natural \(T_0\)-ultra-quasi metric on \(X\) defined by \(d(x,y)= 0\) if \(x\leq y\), and \(d(x,y)= 1\) otherwise, then \(\nu(X,d)\) coincides with the Dedekind-MacNeille completion of \(X\) and \(\nu(X,d)\subset\varepsilon(X,d)\).
scientific article; zbMATH DE number 6568024
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