"multi-dimensional" analogy to linear orders?

[madaco]

(If this post is tl;dr , the lines that just have a few underscores on them and nothing else are places that I think could be convenient to stop reading at, and I would appreciate getting a response to any portion of this post.)

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 an order and the reversed version of that order are counted as being isomorphic [edit : actually I don't think this is true. omega_0 and the reversed version of it are not isomorphic in this. I think it preserves everything about the total orders then.].


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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.)

[madaco]

by the '("preorder" might be replaced with a slightly stronger requirement? I am not sure.)' I think I mean that it might make sense to require it be a total preorder.

The ones constructed from sets of points points in a finite dimensional vector space with the usual inner product all have this property, and these are the main examples I've been considering.

However, it might also be interesting to consider things like infinite binary trees with this setup (though that might work better with not requiring the preorders to be total? I think it would work either way though.).

With these collections of preorders, I think it would make sense to define "the set of points on the line between the points x and y" as, the set of all points z such that for every preorder in the collection, if x is equivalent to y in that preorder, then z is equivalent to y in the preorder as well".

In the ones derived from the vector spaces, this results in the same lines as would be expected, and I think it seems like a reasonable way to define "lines" in this context. However, "z is on the line through x and y" doesn't imply "y is on the line through x and z" for all of these pairs of sets and preorders, if lines are defined this way.

It's true for nice sets of preorders, but not for all of them. e.g. if there is no preorder in the set where x and y are equivalent, then all the points are "on the line through x and y" , but there might be some point z which is equivalent to x in some preorder from the set. Then, the line through x and z would not contain y, as y is not equivalent to x in any of the preorders, so it isn't equivalent in the preorder where x and z are equivalent, but z is on the line through x and y, because all the points are.

This would make "lines" kind of weird if the collection of preorders isn't nice.

So, I don't know if that definition is good.

It seems to me like many properties about how things are arranged relative to each other could be expressed in terms of these sets of preorders (or sets of total preorders).

Like, "is there a separating plane between these two sets" would correspond to "is there at least one of the preorders such that every point in one of the sets is strictly less than every point in the other set, with regards to that preorder"

etc.

With only one preorder, the requirements force it to be a total order, not just a (total?) preorder, and then it is just, whether one of the sets is entirely after the other.