Partially ordered metric spaces produced by T0-quasi-metrics

We continue our investigation on those (partially) ordered metric spaces (X,m,≤)(X,m,≤) for which there exists a T0T0-quasi-metric d on X   such that sup⁡{d,d−1}=msup⁡{d,d−1}=m and for any x,y∈Xx,y∈X we have that x≤yx≤y if and only if d(x,y)=0d(x,y)=0. In particular such partially ordered metric spaces are determined (in the sense of Nachbin) by a T0T0-quasi-uniformity with a countable base.We show that the compatibility conditions obtained in an earlier study between partial order and metric on the investigated spaces X become more transparent when additional compatibility conditions between metric and partial order with appropriate algebraic operations on X are assumed to hold.