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Provided that the notion of "$\mbox{spacelike}$"-ness (of an interval) is symmetric: $$\text{spacelike}( \, x - y \, ) \Longleftrightarrow \text{spacelike}( \, y - x \, ),$$ then for any set $X$ (of sufficiently many elements) the set $X^2$ (of all pairs, regardless of order, of not necessarily distinct elements of $X$) may be partitioned into three disjoint and generally non-empty subsets

  • of pairs containing the same element twice: $I_X := \{ x \in X: (x x) \}$,
  • of pairs of distinct elements whose interval is (called) "$\text{spacelike}$": $S_X := \{ x \in X \, \& \, y \in X \, \& \, x \ne y \, \& \, \text{spacelike}( x - y ): (x y) \}$, and
  • of all remaining pairs $K_X := X^2 \backslash (I_X \cup S_X)$.

Provided further that the notion of "$\mbox{lightlike}$"-ness (of an interval) is symmetric as well: $$\text{lightlike}( \, x - y \, ) \Longleftrightarrow \text{lightlike}( \, y - x \, ),$$ and given a suitable set $X$ (of sufficiently many elements) and a suitable set $S_X$ satisfying $X^2 \cap S_X = S_X$, how would the corresponding set $K_X$ be partitioned further into two disjoint and generally non-empty subsets

  • of pairs of distinct elements whose interval is (called) "$\text{lightlike}$": $L_X := \{ x \in X \, \& \, y \in X \, \& \, x \ne y \, \& \, \text{lightlike}( x - y ): (x y) \}$, and
  • of all remaining pairs $T_X := K_X \backslash L_X$ ?

Edit

The wording of this question (apart from formatting issues) as it presently stands appears not adequate to the title. (Helpful responses to it have been received nevertheless, which I try to incorporate in going forward.)

In trying to improve the detailed wording, what would need to be considered first is (put roughly, for the time being, as I've come to consider it only recently):

(1) Whether and how the topological notion of "boundary" can be suitably generalized to the context of sets of pairs such as $X^2$ and $S_X$, and

(2) Whether, given a particular set $X$, the predicate "$spacelike()$" in the definition of set $S_X$ implies certain relations to a corresponding set $K_X$ which I did not state explicitly above; such as the absence of "impossible figures" wrt. membership of certain pairs in $S_X$ or $K_X$.

I plan to defer editing (apart from possible formatting) until these preliminary questions have been expressed more adequatly elsewhere.

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Comment to the question(v1): The notation used for sets is in many places far from standard, see e.g. this Wikipedia page. –  Qmechanic Sep 18 '12 at 20:55
    
In main body of your question you are effectively asking that if we are given two symmetric relations $S$ and $L$ on a set $X$, and if set of pairs of points which are $S$- related to each other are known, then is it possible to find set of pairs which are $L$-related to each other. –  user10001 Sep 19 '12 at 1:32
    
I wonder if the mathematical-physics tag might be appropriate for this sort of question? –  David Z Sep 19 '12 at 2:30
    
@Qmechanic: Based on the standard you recommend, how would you express for instance "set of pairs (regardless of order) of distinct elements of set $X$", please? –  user12262 Sep 19 '12 at 15:08
1  
@user12262: If you would like, I could edit your question with standard notation. You can always roll back if you don't like the edits. Or improve further yourself. –  Qmechanic Sep 19 '12 at 16:59
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1 Answer 1

I think you can do this only if the set $X$ is a topological space.

For any point $x\in X$, construct the set $C(x)\subset X$ of points $y'\in X$ so that $(x,y')\in K_X$. For this construction to work, each $C(x)\cup\{x\}$ has to be a closed set in $X$. Denote by $\partial C(x)$ the boundary of $C(x)$.

Then,

$$L_X=\{(x,y)\in X\times X|y\in \partial C(x)\}$$

and

$$T_X=\{(x,y)\in X\times X|y\in \text{int } C(x)\}.$$

You can do an equivalent construction starting with $S_X$ instead of $K_X$.


The intuition behind this construction is that the lightcone at a point $x$ is the boundary of the points which are causally connected with $x$.

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@Cristi Stoica: Very nice and efficient and intuitive! (Meanwhile I've been struggling through trying to cast the similar intuition into overlong logical formulas and constructions à la A.A.Robb ... &) However: my question did not presume set $X$ to be given with a topology! (Sorry for having set the "general-relativity" tag then -- is there a more appropriate one? But I'm really glad you understood the sense of my question anyways.) So: Is there perhaps a suitable topology being induced on set $X$, plainly by the given set $S_X$ ? (The sum of difficulties might come out the same.) –  user12262 Sep 18 '12 at 22:00
    
@user12262: If you don't have the topology, you need to have something else from which to extract the needed information. If we only know it is a set, it is not enough. However, surprisingly many situations allow us to construct a suitable topology. Even for so-called discrete theories this can be done (see e.g. philsci-archive.pitt.edu/4355). If you can come with additional structure on your sets, for example if you can define the distance between $x$ and $y$, even if it is only when they are separated by a spacelike interval, then we can use it to solve the problem. –  Cristi Stoica Sep 19 '12 at 2:22
    
@Cristi Stoica: thanks for your insight and suggestions, FWIW to me in trying to phrase my question more adequatly (cmp. the recent "Edit" of my question). I wish I had a better grasp of (standard) mathematical terminology; to use (or avoid) what's available, and to claim what's not ... Presuming distance (or at least quasi-distance) relations is of course not helpful to the experimentalist who rightly asks "How do we get that?". Also, if some "additional structure" itself is available, then it may not be mandatory to consider any topology which may (or may not) be thereby induced. –  user12262 Sep 19 '12 at 17:14
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