Structural analysis of Petri nets has played an important role in the success of Petri nets for the analysis of discrete-event dynamic systems. Most structural Petri net theory has focused on P- (and T-) semiflows, the left (and right) non-negative integer annullers of the Petri net incidence matrix, which capture invariant laws for markings and cyclic behaviors, while little attention has been devoted to the general case of integer annullers, leading us to our first question: Are there useful invariant laws based on arbitrary integer flows? Moreover, the set of semiflows is typically characterized (in theory and tools) by the minimal semiflows, whose entries are relatively prime and whose support does not contain that of other semiflows. Minimal semiflows form a \(\mathbb {Q}_{\ge 0}\) -generator: any semiflow can be written as a linear combination with positive rational coefficients of minimal semiflows. Then, our second question is: Can we characterize the set of (semi)flows through an \(\mathbb {N}\) -generating set, and do we gain better or additional insights by doing so? To answer these questions, we consider the lattice of integer flows and shed light on the role of the Hilbert and Graver bases of a lattice vs. the notion of minimal flows. Moreover, we use various examples to discuss invariant laws based on flows in the Graver basis, and the role of arbitrary integer flows.