On a number of poisson matrices in Bang-Bang representations for 3 - 3 embeddable matrices

We give a counterexample to the Strong Bang-Bang Conjecture according to which any 3 - 3 embeddable matrix can be expressed as a product of six Poisson matrices. We exhibit a 3 - 3 embeddable matrix which can be expressed as a product of seven but not six Poisson matrices. We show that an embeddable 3 - 3 matrix P with det can be expressed as a product of at most six Poisson matrices and give necessary and sufficient conditions for a 3 - 3 stochastic matrix P with det to be embeddable. For an embeddable 3 - 3 matrix P with det we give a new bound for the number of Poisson matrices in its Bang-Bang representation.