Let there be two sets of points $S$ and $S'$ in $R^d$. $|S| = |S'|$, and for each point $s_i$ in $S$ it exists exactly one corresponding point $s'_i$ in $S'$, such that the ordering of $S$ equals the ordering of $S'$ with respect to any axis of $R^d$.Is there any term that describes this relationship or 'morphism' between $S$ and $S'$?E.g. in $R^2$, if $x_{s_i} < x_{s_j}$ then $x_{s'_i} < x_{s'_j}$, or if $y_{s_i} \ge y_{s_j}$ then $y_{s'_i} \ge y_{s'_j}$, etc.
Term for a correspondence of two point sets regarding their ordering in each dimension
cg.comp geom;terminology;metric spaces
Associate with $S$ and $S'$ the posets $P = (\leq, S)$ and $P' = (\leq, S')$, where $\leq$ is the dominance order in $\mathbb{R}^d$. The function you describe is an isomorphism between $P$ and $P'$. 