After working on several versions of the next element in the structure, I decided to start writing today although some sections are still unclear. this is the first few paragraphs:
According to the field of real numbers, zero is the only element that doesn’t have a multiplicative inverse. In extended R, infinity is added to R however the mechanism that links zero to inf is unclear. Define F_1: {0, inf} → {0, inf} as the multiplicative operation where 0 is mapped to inf, and inf is mapped to 0. Also, define F_2 as a function that produces conventional multiplicative inverses used in R.
F_2: R-{0} → R-{0}
It’s clear that F_2 and F_1 are different. That implies that the multiplication operations are different in these two situations. In other words, if R is extended to include {0} the multiplication operator requires further considerations. The definition and properties of F_2 are clear in R, while the properties of F_1 are not.
Apparently, there is no indication on how to construct F_1. However, a comparison between R and SR can provide some indications. Based on Mink, the ct-axis refers to v=0 and the x-axis pertains to v=inf. Although the physical implication of v=inf is unclear, mathematically the slope of the x-axis refers to v=inf. Mink then provides a mathematical structure that relates the ct and x axes of moving objects. In other words, Mink is all about the connection between zero and inf, and the way they are linked together. The connection as discussed in the previous section uses inversion relative to the unit circle and links intervals [0,1) and (1, inf) (or regions v<c and v>c) to zero and inf. As a result, the connection between zero and inf is extended throughout the entire R. So the extended R doesn’t merely add a new element, inf, to R. It also establishes a new connection between [0,1) and (1, inf) beyond the conventional multiplication operation defined in R.
UPDATE 1:
Adding another dim to R establishes new connections between [0,1) and (1, inf). This had been discussed before. One of the earliest insights into the concept of inf was that it was a dimension. However, this idea was unclear at the time, because inf is one point and it was unclear how adding one point to some space would add another dimension to that. This concept is becoming clearer now. For example, when inf is added to R (in stereographic projection), the line of real numbers turns into a circle. The circle is still a 1-d object yet defined in a 2d space. In projective geometry, the point at inf requires parallel lines spaced along the z-axis. Again adding a new element, inf, adds a new dimension to R. However there is a huge difference between this “dimension” and conventional dimensions used in math. R can be extended to R^2 and R^3 and so on. But on the contrary, when inf is added, a type of mechanism is added to the space. It’s not just one additional dimension, instead, it’s an n+1 dim space that contains some mechanism. For example, in projective geometry adding the z-axis doesn’t just add another dim, it constructs a space where a mechanism that pushes R toward the point at inf is included. Spatial axes resemble R^n. Mathematically one can add new dim to some space. A particle can move in a 1d or 2d spatial space. However, when inf is added, the new space is accompanied by a mechanism.
About the first example, when R floats in n+1 space, new connections between [0,1) and (1, inf) can be constructed. When R is placed in 2d more properties of R (as a 1d space) can be attained compared to when R is contained in 1d space. This process can go on. R, a 1d space, can be placed in n+1 space and add new properties to the R. R is still 1d but has new properties.
This will have implications in physical spaces.