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This is the generalization of a question Is a simple graph the "sum" of a partial order and its dual? Nik Weaver found a counterexample in a very nice, complete (and instantaneous!) answer,...

A "$n$-order matrix" $T\in M_n(\mathbb F_2)$ is a matrix such that there exists a partial ordered relation $\leq_T\subset [1,n]^2$ such that : $T_{ij}=1\Leftrightarrow i\leq_T j$ (where $T_{ij}$ is ...