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Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of the original dot. The countable sets of such dots have some property, let call it "class".

There are operations on the sets that keep the class unchanged:

  • Moving finite number of dots, as well as merging and splitting finite number of dots (summing and subtracting the weights respectively) does not change the class. Moving an infinite set that is bounded from left and right also does not change the class.

  • Reflecting any subset of a set against zero (that is moving the points to the opposite side of zero) does not change the class. This includes the case with splitting and merging points in the process.

  • If a set has a point of symmetry, it can be as a whole moved either right or left, this keeps the class unchanged. This is also the case if the subsets left and right of the dot do not exactly match but have the same class.

  • If there are two subsets of the same class, and located at the sime side of zero, one can be moved right, the other can be moved left the same distance.

  • If there are two subsets of the same class, the weight of one can be multiplied by a factor, the weight of the other divided by the same factor, the class of the whole set will not change. One can be scaled up, the other can be scaled down the same factor (the distances of all dots from the zero scaled), the class of the whole set will not change.

  • A set of points located at the distances $x_i$ from zero has the same class as the set of points located at $1/x_i$ with the same weights.

  • Any operation that does not change the class applied to a subset also does not change the class of the whole set.

Given the above rules, what are possible canonical forms for the sets? Particularly, can any such set be transformed to a set of dots located at non-negative integers with keeping the class intact?

If there an easier way to define such classes with simpler rules?

Especially it would be great if there were a formula based on the coordinates and wei9ghts of the dot that would indicate whether two sets are of the same class.

Asked at Math.SE, proposed a bounty but no attention attracted.

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    $\begingroup$ If the set has an accumulation point, it seems that applying the rules a finite number of times, doesn't allow undoing that; so that the set will not be equivalent to one based on on the integers. $\endgroup$ – Yaakov Baruch Apr 11 '16 at 10:55
  • $\begingroup$ @Yaakov Baruch what about cases without accumulation point? $\endgroup$ – Anixx Apr 11 '16 at 11:46
  • $\begingroup$ @Yaakov Baruch I have added a rule for accumulation points, second from the end. $\endgroup$ – Anixx Apr 11 '16 at 13:47
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    $\begingroup$ How intriguing. While we think about it, could you please explain briefly why this gadget is of interest to you? $\endgroup$ – Vidit Nanda Apr 11 '16 at 14:25
  • $\begingroup$ Also, just to see if I get the rules: if we have "all positive integers, zero and all halved negative integers", how would you find an equivalent representative with only positive dots? Let's say all weights are 1. $\endgroup$ – Vidit Nanda Apr 11 '16 at 14:29

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