Asymmetric norms given by symmetrisation and specialisation order (Q1637105)

A function \(d:X\times X\to [0,\infty)\) of a set \(X\) is called a \(T_0\)-quasi-metric if the following conditions hold for all \(x,y,z\in X\): \[ d(x,x)=0, \] \[ d(x,z)\leq d(x,y)+d(y,z), \] \[ d(x,y)=0=d(y,x) \text{ implies that }x=y. \] The authors in this paper continue the investigation between \(T_0\)-quasi-metric spaces and partially ordered metric spaces. In Section \(2\), they show that it is possible to set up a Galois connection between these two classes of spaces (see Proposition \(2.2(c)\)). Furthermore, in Section \(3\), the authors establish a relationship between nonexpansive maps between \(T_0\)-quasi-metric spaces and nonexpansive and increasing maps between partially ordered metric spaces (see Propositions \(3.1\) and \(3.2\)). Finally, in the last section (Section \(4\)), they derive a representation theorem for injective asymmetrically normed spaces. This answers a question that was left open in [\textit{J. Conradie} et al., Topology Appl. 231, 92--112 (2017; Zbl 1387.46050)].