# Term for a correspondence of two point sets regarding their ordering in each dimension

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.

• Maybe a general term would be 'order isomorphic'? (en.wikipedia.org/wiki/Order_isomorphism) -- but there might be something more specific? – 0__ Jun 29 '11 at 1:09
• A pointwise order-preserving function or pointwise monotone function. – Dave Clarke Jun 29 '11 at 6:47
• I think that simply monotone suffices in this case, because the function maps $S\to S'$. – John Moeller Jun 29 '11 at 14:50

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 $P'$.