Asymmetric norms given by symmetrisation and specialisation order

In this paper we continue the investigations of the relationship between T0-quasi-metric spaces and partially ordered metric spaces. Among other things, we establish an equivalence of categories between so-called maximal T0-quasi-metric spaces and partially ordered metric spaces produced by T0-quasi-metrics. In the linear context we give geometric interpretations of the obtained results. In particular, we also derive a representation theorem for injective asymmetrically normed spaces.