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Since there is an isomorphism between split-complex numbers and $\mathbb{R}^2$ with element-wise operations, of the following form $a + bj \leftrightarrow (a - b, a + b)$, one can think about a split-complex number as of expression of the form $a\pm b$, that is, the split-complex unity $j$ representing the $\pm$ sign and can be written down as $\pm1$.

Thus, a pair $(u,v)$ represents a split-complex number $\frac{v+u}2+j\frac{v-u}2$, fo instance $(-1,1)=0\pm1$ represents $j$ while any interval which has an end at zero is a zero divisor. Particularly, we can see that sign function takes 9 distinct values at split-complex numbers, with all intervals with both ends positive being positive split-complex numbers.

In other words, the middle of the interval $(u,v)$ is the real part, while the split-imaginary part represents deviance. The change of sign of split-imaginary part swaps the interval's ends.

But if we consider the intervals $(u,v)$ such that $u,v\in\overline{\mathbb{R}}$, there are some intervals that cannot be represented in $a+bj$ form. For instance, $(0,\infty)$, $(2, -\infty)$, $(-\infty,\infty)$, etc.

This way, we can extend the set of split-complex numbers and define operations on this extended set.

So, my question is, what would be the properties of such compactification? Was it ever described?

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  • $\begingroup$ "we can extend the set of split-complex numbers and define operations on this extended set." How do you define the operations? $\endgroup$
    – Wojowu
    Commented Dec 14, 2021 at 11:06
  • $\begingroup$ @Wojowu well, due to isomorphism, $f(u,v)=(f(u),f(v))$, so when $f$ is defined on $\overline{\mathbb{R}}$, it is defined on extended split-complex numbers as well. $\endgroup$
    – Anixx
    Commented Dec 14, 2021 at 11:12
  • $\begingroup$ One problem is that you seem to be representing a quotient of $\mathbb R^2$, not $\mathbb R^2$ itself. That is, your pair notation has no way of distinguishing $+j = \langle{-1}, 1\rangle$ from $-j = \langle1, -1\rangle$, although they are distinct elements of $\mathbb R^2$. For example, should the sum of the interval (not the ordered pair) $(-1, 1)$ with itself be $(0, 0) = \emptyset$ or $(-2, 2)$? $\endgroup$
    – LSpice
    Commented Dec 14, 2021 at 11:37
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    $\begingroup$ The interval $(-1, 1)$ stands for both $+j$ and $-j$, so its sum with itself could be $+2j$ or $-2j$, both represented by $(-2, 2)$, or $0$, represented by $(0, 0)$. $\endgroup$
    – LSpice
    Commented Dec 14, 2021 at 11:42
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    $\begingroup$ So what does $(1, -1)$ mean as an interval? And, again, if it is just a pair, then what do you gain by writing an element of $\mathbb R^2$ in one basis $\{(1, 0), (0, 1)\}$ rather than another $\{(1/2, -1/2), (1/2, 1/2)\}$? $\endgroup$
    – LSpice
    Commented Dec 14, 2021 at 11:49

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There is a compactification of the split-complex numbers. I will denote the split-complex numbers as $\mathbb R^2$ as you suggested. The consequences of such a compactification are described here:

https://en.wikipedia.org/wiki/Laguerre_transformations#Other_number_systems_and_the_parallel_postulate

and here:

https://en.wikipedia.org/wiki/Line_coordinates.

This compactification deserves to be called the projective line over $\mathbb R^2$. A book by Isaac Yaglom called Complex Numbers in Geometry investigates it, as well as projective lines over the dual numbers and complex numbers. The general definition of a projective line over a ring is what's needed here.

The elements of the projective line over $\mathbb R^2$ represent oriented lines in the hyperbolic plane. Analogously, the projective line over the dual numbers represents oriented lines in the Euclidean plane. Over the complex numbers, oriented lines in the elliptic plane. Notice the three classic non-Euclidean geometries {hyperbolic, Euclidean, elliptic} correspond to the three planar number systems {split-complex, dual, complex}.

[Edit] I've noticed that your examples involve things like $(-\infty, 2)$, which suggests you're treating $\infty$ differently from $-\infty$. This ends up producing a different compactification from the one I suggested. In my opinion, the interesting geometry of the split-complex numbers cannot be reduced to the operations $\{+,-,\times,/\}$ over it, but rather you have to consider the involution $(a,b)^* := (b,a)$ as well. Maybe you can find some interesting phenomena there. You might want to also check if operations on the extended real numbers can be combined with the involution to produce interesting phenomena.

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  • $\begingroup$ The thing is, different basises bring different compactifications. In one basis we add a family of oriented lines, parallel and pependiclar to the real line, in the other we add a family of oriented diagonal lines. $\endgroup$
    – Anixx
    Commented Dec 14, 2021 at 13:31
  • $\begingroup$ I would argue that the compactification with lines parallel and perpendicular to the real line is more natural for the complex numbers, and the compactification by adding diagonal lines is more natural for hyperbolic numbers. $\endgroup$
    – Anixx
    Commented Dec 14, 2021 at 13:32
  • $\begingroup$ The compactification with diagnal lines allows to easily generalize functions to this extended set in the diagonal basis. $\endgroup$
    – Anixx
    Commented Dec 14, 2021 at 13:42
  • $\begingroup$ @Anixx I don't understand what definition of compactification you're using. I'm using the definition of a projective line over a ring, which is basis independent $\endgroup$
    – wlad
    Commented Dec 14, 2021 at 15:11
  • $\begingroup$ @Anixx en.wikipedia.org/wiki/Projective_line_over_a_ring $\endgroup$
    – wlad
    Commented Dec 14, 2021 at 15:15

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