Let's take $M$ be a $n\times n$ matrix whose entries are $0$ or $1$. (then we can call it the *characteristic matrix* of any relation  $R_M\subset \left\{a_1,...,a_n\right\}^2$, such that $M_{ij}=1$ iff $(a_i,a_j)\in R$)
 
$\mathcal C(M)$ is defined to be the matrix that columns are the same than that of $M$ but ordered according to the *lexicographic order*. 

(i.e  there exists $M_{\mathcal C}$ permutation matrix such that $\mathcal C(M).M_{\mathcal C}=M$ and such that $[2^n,2^{n-1},...,1]\mathcal C(M)(e_i-e_j)>0$, for any $i,j\in [1,n]$ , where $e_i$ is the column with $0$ everywhere except on the $i$-th projection).

 We also define $\mathcal R(M):=(-C(-M^t))^t$ 

(i.e $\mathcal R(M)$ is obtained by ordering rows of $M$ in the **decreasing** *lexicographic order*.) And we now consider $\mathcal L=\mathcal R\,o_,\mathcal C$ and $\mathcal L^2=\mathcal L\,o\,\mathcal L$, that any fixed point is called "cycle matrix". Then a relation that characteristic  matrix is a cycle matrix will be said  *cycle-indexed*)

>Is it true that for any matrix $M$ there is an integer $k$ such that $\mathcal L^{k+2}(M)=\mathcal L^k(M)$

In other words, is any binary relation on a set, "cycle-indexable"?

-Note that one can chose an ordinal indexation of any set equipped with a binary relation, such that the following consideration are still relevant for infinity...

I think the answer to the question is yes and that it may not be to hard to prove, but my biggest interest, is for the matrices  that are both **cycle matrix** and **up-tridiagonal** with $1$ on the diagonal. If $M$ is such a matrix, and if $\mathcal L(M)$ is also up-tridiagonal, I will call it  an *almost ordered matrice*. And the reason for this is that characteristic matrix of partial ordered relation are, up to a fine indexation of the partial ordered set, *almost ordered matrix*. Meaning that for any "order matrix" $M$, you can find $P$ permutation matrix such that $P^t.M.P$ is an *almost order*.  And there is more : for any permutation matrix $P$ and $Q$, we can say that $\mathcal L(P.M.Q) $ is a tridiagonal matrix, it is quite easy to see, but I find this quite nice anyway... There is already plenty of questions that are coming to me and I will  ask some specifics ones in other topics. Just to give an example : if you remove  elements of a partial ordered "cycle indexed" set, then you trivially get a partial ordered set,  *but* it is quite amazing to constate  that the induced indexation is still a cycle-indexation ! **[edit : this is only true fore some éléments of the ordered set, like the one with top index** :maybe it is true in a lattice  if the removed element is a meet-irreducible... but I've got to check, however characterizing these elements that when you remove them, you still get a cycle, seems interesting)] this is far from being true for general matrices/relations!) I also suspect that this very simple $\mathcal L$ could have applications to lattices and to graph theory (cycle matrix of symmetric matrix is not always symmetric "but" ...?...), complexity (how many cycle indexation compare to $(n!)^2$...) and also other areas (arithmetic progression seem concerned too...), but before going into this work, and in order to get a *direction* and an evaluation of the application and the "meaning" of this "almost transitivity" that would have an anti-symmetric and reflexive relation that would not  be a "partial" but an "almost order", I'm going to ask a question that is a bit general :

>Is there a nice characterization of tridiagonaly cycle-indexed relations ("almost order") ?


I would be pleased with any answer that would give some interesting properties even if not a proper characterization**, and I also would like to know if this subject has already been treated in literature...


  


**If there is not such a "nice" characterization, maybe there is useful properties that we can use in a smart=relevant generalization of partial orders. For if there is no "nice" characterization, it could  even be more exciting, because that would kind of suggest that it is potentially a  mean-full or self-mean concept...