Term for a correspondence of two point sets regarding their ordering in each dimension Let there be two sets of points $S$ and $S&#39;$ in $R^d$. $|S| = |S&#39;|$, and for each point $s_i$ in $S$ it exists exactly one corresponding point $s&#39;_i$ in $S&#39;$, such that the ordering of $S$ equals the ordering of $S&#39;$ with respect to any axis of $R^d$.

Is there any term that describes this relationship or 'morphism' between $S$ and $S&#39;$?

E.g. in $R^2$, if $x_{s_i} &lt; x_{s_j}$ then $x_{s&#39;_i} &lt; x_{s&#39;_j}$, or if $y_{s_i} \ge y_{s_j}$ then $y_{s&#39;_i} \ge y_{s&#39;_j}$, etc.
Associate with $S$ and $S&#39;$ the posets $P = (\leq, S)$ and $P&#39; = (\leq, S&#39;)$, where $\leq$ is the dominance order in $\mathbb{R}^d$. The function you describe is an isomorphism between $P$ and $P&#39;$.