Calculus without limits

# Noncommutative Space

Coordinates are function values. On $\mathbb{R}^2-$ the two functions $f(x,y) = x$ and $g(x,y) = y$ , allow to determine every point in the plane. In the continuum coordinates are commutative: $\{ f=c,g=d \}$ is the same than $\{ g=d, f=c \}$. This commutativity also holds in Riemannmian manifold settings. In the discrete this is no more true. Lets for simplicity assume we want to pinpoint the point (0,0) using the function f(x,y)=0, g(x,y) = 0. If [4]=(++,+-,-+,–) are the possible sign outcomes of the functions f,g, We have seen how to build a frame work where f=c, g=d are submanifolds. Here is an illustration for a very simple 2 dimensional manifold situation (a discrete flat plane, an open 2-manifold for which the curvature is zero everywhere. The first picture shows the level curve of f(x,y) = x, the second the level curve of f(x,y)=y (notice that they are manifolds! ). The third shows the circle (constant distance to a point). For the last one, I chose a random function f(x,y) and deform it. Now what happens if we chose a procedure to determine the level points {f==c, g=y=d} is that either, we have {f-c=0, g-c=0} being different from {c-f=0, g-c=0} or {c-f=0, c-g=0} or then different from {g-d=0, f-c=0}. Order and sign can matter because we have to make a choice when choosing the 2 dimensional partition complex on [4], the set of possible sign values. There is up to ismorphism just one partition of 4 which qualifies as a 2 dimensional complex and that is the kite graph K(1,1,2) belonging go the partition 4=1+1+2. This is the only partition of 4 which has 3 sumands.