Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two 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 the set is an infinite grid (has equidistant equal weight points), scaling the set up by a positive factor $a$ and multiplying all the weights by the same factor keeps the class unchanged. Scaling the set down and dividing the weights by the same factor also leaves the class unchanged.

* 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. One can be scaled up, the other can be scaled down the same factor, the class of the whole set will not change.

* 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.

* 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?

Asked [at Math.SE][1], proposed a bounty but no attention attracted.


  [1]: http://math.stackexchange.com/questions/1696966/classifying-countable-sets-of-weighted-dots-on-a-real-line