I would like to know if the following conjecture is correct and/or already known. Do you have any ideas for a demonstration or counter-example? Thank you.
Let $X$ be a finite set and $(P_i)_{i\in N}$ a finite sequence of linear orders of $X$ such that $$\forall x,y\in X,\quad |\{i\in N \mid xP_iy\}|=|\{i\in N \mid yP_ix\}|.$$ For each linear order $P$, we define $\bar{P}$ as follows: $\ a\bar{P}b$ if and only if $bPa.$
Conjecture: There exists a partition $\{N_0,N_1\}$ of $N$ and a bijection $b:N_0\rightarrow N_1$ sur that for each $i\in N_0$, $P_{b(i)}=\bar{P_i}$.
I just found a counterexample: $P_1=(a,b,c,d)$, $P_2=(c,b,a,d)$, $P_3=(d,c,a,b)$, $P_4=(d,b,a,c)$.