A Disconnected 1-Safe Petri Net Whose Reachability Tree Is Homomorphic to a Complete Boolean Lattice

Petri nets generating all the 2ⁿ binary n-vectors as their marking vectors are not only of theoretical interest but also are of practical importance. In this note, we demonstrate the existence of a disconnected 1-safe Petri net whose reachability tree is homomorphic to the n-dimensional complete lattice L_n. This makes the problem of characterizing the 1-safe Petri nets that generate all the binary n-vectors as marking vectors exactly once appear more intricate.