I thought about ways to generalize the equality relation to more than two numbers. If we say that three numbers are equal, we actually mean that they are pairwise equal, i.e. it turns into three equality relations for two numbers.
However, the collinearity relation naturally applies to three objects, points in Euclidean space with two or more dimensions. Three points always define a plane, thus we can restrict ourselves to the 2D case.
In the same way, there is coplanarity of 4 points in 3D space.
Now, to explain myself better, let's go 'from the top down':
- We say that 4 points are coplanar if they lie on the same plane. 3 points always lie on the same plane.
- We say that 3 points are collinear if they lie on the same (straight) line. 2 points always lie on the same line.
- We say that 2 numbers (2 points on a straight line) are equal if they 'lie on the same point'. 1 number is always equal to itself.
So, can we say that collinearity and coplanarity are (in some way) generalizations of the equality relation?
I think there can be certainly other generalizations, which is why I added 'in some way'.
A reference to some authorative source would be appreciated as well.
Yes, this is a generalization, and I actually like it quite a lot. The only downside is that it doesn't make sense to apply your notion to three real numbers (or else every three real numbers are equal since they span something not three dimensional).
A further generalization of this is called matroids, which you might find interesting. They move away from geometry into more abstract notions. They can also be used in number theory to say meaningful things like whether numbers are "algebraically independent" or not.