Splitting ultra-metrics by \(T_{0}\)-ultra-quasi-metrics (Q1744597)

By an ultra-quasi-pseudometric on a nonempty set \(X\), we mean a function \(d:X\times X\to [0,\infty)\) satisfying the following conditions (a) \(d(x,x)=0\) whenever \(x\in X\) and (b) \(d(x,y)\leq\max\{d(x,z),d(z,y)\}\) whenever \(x,y,z\in X\). If in addition \(d\) satisfies (c) \(d(x,y)=0=d(y,x)\) implies that \(x=y\) whenever \(x,y\in X\), we say that \(d\) is a \(T_0\)-ultra-quasi-metric on \(X\) and we call the pair \((X,d)\) a \(T_0\)-ultra-quasi-metric space. Given an ultra-metric space \((X,m)\) we call an ultra-quasi-pseudometric \(d\) on \(X\) \(m\)-splitting provided that \(d^s=d\vee d^{-1}=m\). An \(m\)-splitting ultra-quasi-pseudometric \(d\) on an ultra-metric space \((X,m)\) is called \(U(X)\)-minimally \(m\)-splitting provided that whenever \(e\) is an ultra-quasi-pseudometric on \(X\) with \(e\leq d\) and \(e^s=m\), then \(e=d\). If \((X,m)\) is an ultra-metric space equipped with a partial order \(\leq\) on \(X\), then we say that \((X,m,\leq)\) is a partially ordered ultra-metric space. We say that \((X,m,\leq)\) is produced by a quasi-pseudometric \(d\) on \(X\) if \((1)\; d\) is \(m\)-splitting and \((2)\; \leq\) is the specialization (pre)order on \(d\). In this article under review, the authors study (partially) ordered ultra-metric spaces \((X,m,\leq)\) such that there exists a \(T_0\)-ultra-quasi-metric \(d\) on \(X\) the specialization order of which is equal to \(\leq\) and \(d^s=m\). They show amongst other things that every ultra-metric space \((X,m)\) with an \(m\)-splitting \(T_0\)-ultra-quasi-metric has a \(U(X)\)-minimal \(m\)-splitting \(T_0\)-ultra-quasi-metric (see Proposition \(1\)). The authors present two open problems: one on producibility (Problem \(1\)) and the other one on the existence of linear orders on an ultra-metric space satisfying a certain condition (Problem \(2\)). We note that a partial solution to Problem 2 is presented. Moreover, several examples are presented by the authors to demonstrate their results.