I would like your help to find some necessary and sufficient conditions for equivalence between the outputs of two operators. Could you help me with the tags? I don't know which branch of math my question belongs to (if to any).
Consider the ordered quadruplet $(a,b,c,d)\in \mathbb{R}^ 4$ and define two ordered 9-tuplets $$ f(a,b,c,d)\equiv (-a, -b, a, b, -c, -d, c, d, 0) $$ $$ g(a,b,c,d)\equiv (b-a, -b, a-b, b, d-c, -d, c-d, d,0) $$
Define the operator $\pi_f(a,b,c,d)$ giving
1) the position in the original 9-tuple of the elements of $f(a,b,c,d)$ when ordered from smallest to largest
2) the relational operator ($<,=$) between the elements of $f(a,b,c,d)$ when ordered from smallest to largest
Define the operator $\pi_g(a,b,c,d)$ giving
1) the position in the original 9-tuple of the elements of $g(a,b,c,d)$ when ordered from smallest to largest
2) the relational operator ($<,=$) between the elements of $g(a,b,c,d)$ when ordered from smallest to largest
For example, take $(a,b,c,d)\equiv(-2, 3, -9, -7)$ so that $$ f(-2, 3, -9, -7)=(2, -3, -2, 3, 9, 7, -9, -7, 0) $$ $$ g(-2, 3, -9, -7)=(5, -3, -5, 3,2, 7, -2, -7, 0) $$ When we order from smallest to largest $f(-2, 3, -9, -7)$ we get $$ (-9, -7, -3, -2, 0, 2, 3, 7, 9) $$ When we order from smallest to largest $g(-2, 3, -9, -7)$ we get $$ (-7, -5, -3, -2, 0, 2, 3, 5, 7) $$ Hence, $$ \pi_f(-2, 3, -9, -7)= \Big \{(7, 8, 2, 3, 9, 1, 4, 6, 5), (<, <, <, <, <, <, <, <) \Big\} $$ and $$ \pi_g(-2, 3, -9, -7)= \Big \{(8, 3, 2, 7, 9, 5, 4, 1, 6), (<, <, <, <, <, <, <, <) \Big\} $$
When two elements of $f$ (or $g$) are equal we assume some convention, e.g., we put first the element with the smaller position in $f$ (or $g$).
Question:
Take two quadruplets $(a,b,c,d)$ and $(a',b',c',d')$.
Is it possible to find necessary and sufficient conditions on $(a,b,c,d),(a',b',c',d')$ such that $$ \begin{cases} \pi_f(a,b,c,d)=\pi_f(a',b',c',d')\\ \pi_g(a,b,c,d)=\pi_g(a',b',c',d') \end{cases}$$ In other words, are the some relations between $(a,b,c,d)$ and $(a',b',c',d')$ that imply and are implied by $\pi_f(a,b,c,d)=\pi_f(a',b',c',d')$ and $\pi_g(a,b,c,d)=\pi_g(a',b',c',d')$?
Following a suggestion below: It is also OK if you help me with necessary and sufficient conditions on $\mathbb{R}^2$ (e.g., just considering $(a,b)$ and deleting $(c,d)$). In addition to that, we can start with restricting the attention to $(a,b), (a',b')$ such that $a\neq b\neq 0$ and $a'\neq b'\neq 0$, so that the second outputs of $\pi_g, \pi_f$ do not matter.
As I'm not a mathematician, I really look forward to get some help to formalise these things.