I was trying to either find an existing description of, or come up with, something analogous to linear orders / (non-strict) total orders, but which works for things in more than one dimension (with the goal of after that, finding a notion of something like a scattered order for such things), and I came up with an idea for it, but I am curious if anyone knows of, or will come up with, a better one.

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Here is what I came up with:

Instead of having a set of points along with a total order on that set, have a set of points along with a set of pre-orders on that set, such that for every pair of distinct points from the set x_1 , x_2, one of the pre-orders associated with the set has x_1 < x_2 . (or possibly, either has x_1 < x_2 , or has x_2 < x_1 (not both though))

("preorder" might be replaced with a slightly stronger requirement? I am not sure.)

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Two of these pairs (of a set of "points" and a set of pre-orders on the set having the required property) are isomorphic if there exists a pair of bijections, where the first one is between the two sets of points, and the second one is between the sets of preorders on the points, such that for any pair of points x,y from the first pair, and for any preorder from the first pair, the preorder has x <= y (where <= is the preorder in question) iff f(x) g(<=) f(y)

( g(<=) is the preorder from the second pair that the one in the first pair gets mapped to. I don't know what a good way to notate this is. It looks weird to put a symbol like <= as the argument to a function? )

When one of these pairs only has one preorder in the set of preorders part, it is equivalent to a (non-strict) total order, with the only thing lost being that

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If you define one one such pair based on position in a (finite dimensional) vector space in a particular way, I have a few results about when two of these are isomorphic.

Let the points be from the vector space V = R^n with the usual dot product.

For v in V , let L(v) be the preorder defined as (x L(v) y) iff ( (x . v) <= (y . v) ) .

For some subset S of V, pick the pair defined as : (S , { L(v) | v in V } )

If two such pairs defined in such a way from sets S and S' have S and S' be translations, rotations, or scalings of eachother, then the corresponding pairs are isomorphic.

If all the points in S except for 1 are in a 1 dimensional subspace of V, then the pre-orders include:

1) for each of the elements of S other than the 1, a preorder which keeps all of those in the same total order (or reversed) , and has the extra point equivalent to that one

2) one where all of the points except for the 1 are equivalent, and the extra point is either less than or equal to all the rest

3) one where all the points are equivalent (whether to exclude this in the definition or not I don't know. It seems inconsequential.)

and possibly

4) one where all the points but the one are in their order (or the reverse of it) and the extra point is greater than all of them, and/or one where it is less than all of them

5) possibly more that I haven't thought of, but I think this is probably all of them.

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I don't know if this is a reasonable idea to define. Any and all feedback is appreciated.

Do you have advice for when it makes sense to define a type of structure? Does "I am going to try to come up with a type of structure that captures this idea" make sense as a motivation, or do the definitions for types of structures generally come around gradually, bubbling up around work people are already doing, and then it just gets formalized later?

In addition to feedback on the idea, I am also interested in feedback on the presentation of the idea.

(also, if this idea isn't something I ought to immediately abandon, would it make sense to give these pairs a name (assuming they don't already have one given to them by someone else before I thought of them I mean)? In the text document I have I have been calling them a "smidge" for lack of a good name.)