<p>Associate with $S$ and $S&#39;$ the posets $P = (\leq, S)$ and $P&#39; = (\leq, S&#39;)$, where $\leq$ is the <a href="http://mathworld.wolfram.com/Dominance.html" rel="nofollow">dominance order</a> in $\mathbb{R}^d$. The function you describe is an <a href="http://mathworld.wolfram.com/IsomorphicPosets.html" rel="nofollow">isomorphism</a> between $P$ and $P&#39;$. </p>

CC BY-SA 3.0
2011-07-01T20:29:25.460Z
2011-07-01T20:29:25.460Z
Sasho Nikolov
<p>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 $R^d$.</p>

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

<p>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.</p>

CC BY-SA 3.0
2011-06-29T01:09:00.230Z
Maybe a general term would be 'order isomorphic'? (http://en.wikipedia.org/wiki/Order_isomorphism) -- but there might be something more specific?
0__
CC BY-SA 3.0
2011-06-29T06:47:39.393Z
A _pointwise order-preserving function_ or _pointwise monotone function_.
Dave Clarke
CC BY-SA 3.0
2011-06-29T14:50:13.013Z
I think that simply _monotone_ suffices in this case, because the function maps $S\to S'$.
Josephine Moeller
<cg.comp-geom><terminology><metric-spaces>
Term for a correspondence of two point sets regarding their ordering in each dimension
0__
# Term for a correspondence of two point sets regarding their ordering in each dimension

(Asked by: username_0 on 2011-06-29)

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.

**Tags:** cg.comp-geom, terminology, metric-spaces

> **Comments:**
> **username_0**: Maybe a general term would be 'order isomorphic'? (http://en.wikipedia.org/wiki/Order_isomorphism) -- but there might be something more specific?  
> **username_1**: A _pointwise order-preserving function_ or _pointwise monotone function_.  
> **username_2**: I think that simply _monotone_ suffices in this case, because the function maps $S\to S'$.  

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## 1 answers

### Answer by username_3 on 2011-07-01. Score: 1 | Accepted answer

Associate with $S$ and $S'$ the posets $P = (\leq, S)$ and $P' = (\leq, S')$, where $\leq$ is the [dominance order](http://mathworld.wolfram.com/Dominance.html) in $\mathbb{R}^d$. The function you describe is an [isomorphism](http://mathworld.wolfram.com/IsomorphicPosets.html) between $P$ and $P'$.


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