N-free extensions of posets. Note on a theorem of P.A.Grillet

Let SN(P) be the poset obtained by adding a dummy vertex on each diagonal edge of the N’s of a finite poset P. We show that SN(SN(P)) is N-free. It follows that this poset is the smallest N-free barycentric subdivision of the diagram of P, poset whose existence was proved by P.A. Grillet. This is also the poset obtained by the algorithm starting with P0 := P and consisting at step m of adding a dummy vertex on a diagonal edge of some N in Pm, proving that the result of this algorithm does not depend upon the particular choice of the diagonal edge chosen at each step. These results are linked to drawing of posets.