Explanation: I want when the order relation $(a\lt b)$ is verified that the result is positive and that if it is not respected, the result is negative, consequently if it is equal, it must be neither one nor the other, i.e. $0$.
In maths it give :
$For\ a < b \ with\ (a,b) \in Z^2 :$
axiom 1 : $ a⨭b = |a+b|;$
axiom 2 : $b⨭a = -( a⨭b) = a⨮b = - |a+b|$;
axiom 3 : $b⨮a = a⨭b$;
axiom 4 : $a⨭a = a⨮a = 0$
(think ⨭as +< and ⨮ as +> or -<)
During my research i find this: Grassmann number : https://en.wikipedia.org/wiki/Grassmann_number
We can see this used in Astrophysics for Supersymmetry (La Supersymétrie; ScienceClic; https://youtu.be/fDeqvjgQ46s?si=wC7DRwl74KG-w-k3&t=356)
$$\theta_a \times \theta_b = - \theta_b \times \theta_a $$ $$\theta_x^2 = - \theta_x^2 = 0$$
→ I think what i wrote in my question is a sort of weird distance, it don't respect triangle inequality but Travel time don't respect it too !
ex :
A shorter path that crosses a mountain and takes 600 minutes.
Vs
A detour through a town that takes 20 minutes to reach the town, then 25 minutes to reach the objective.
https://zupimages.net/up/23/51/8jsi.png
Also a distance can be negative ! : https://fr.wikipedia.org/wiki/Mesure_alg%C3%A9brique (btw sorry, but weirdly not English page for Algebraic measurement)
$$\overline{AB} = AB$$ et $$\overline{BA} = -AB$$
So i will try to reformulate my structure into distance language (but i know it can't be a measure, a measure is positive and must respect the triangle inequality) :
$For\ a < b \ with\ (a,b) \in Z^2 :$
$ a⨭b = |a+b|;$ -> $d(a,b) = |a+b|$
$b⨭a = -( a⨭b) = - |a+b|$; -> $d(b,a) = - |a+b|$
$a⨭a = 0$ -> $d(a,a) = 0$
So do you think it's a distance ? If not, what it could be ?
Ok, if i want this to be more likely a distance maybe i should have define it like this : $For\ a < b \ with\ (a,b) \in Z^2 : $
axiom 1 : $ a⨭b = |a|+|b|;$
axiom 2 : $ b⨭a = -( a⨭b) = - (|a|+|b|)$;
axiom 3 : $ a⨭a = 0$
like this $-2⨭2 = 4 $ (and not 0 like previously, that was an issue) but
and $2⨭-2 = -4$
Like this is more similar to algebraic measurement i show in my question.