On place invariant sets and the rank of the incidence matrix of Petri nets

We discuss the importance of P-invariant analysis of Petri nets and its relationship to the controllability and boundedness of the same. We discuss why a Petri net cannot be fully controllable and bounded simultaneously. It is often mentioned in literature that the rank of the incidence matrix of a Petri net is the difference between the number of places and the number of minimal place invariant sets in the Petri net. We introduce a counterexample in which the above relationship is not true, and then point out the reason for the discrepancy. The discrepancy arises because of the nonnegativity restriction imposed by the definition of the place invariant sets.