We discuss important characteristics of finite generating sets for $$\mathcal {F^{+}}$$F+, the set of all semiflows with non-negative coordinates of a Petri Net. We endeavor to regroup a number of algebraic results dispersed throughout the Petri Nets literature and also to better position the results while considering semirings such as $$\mathbb {N}$$N or $$\mathbb {Q^+}$$Q+ then fields such as $$\mathbb {Q}$$Q. As accurately as possible, we provide a range of new algebraic results on minimal semiflows, minimal supports, and finite minimal generating sets for a given family of semiflows. Minimality of semiflows and of support are critical to develop effective analysis of invariants and behavioral properties of Petri Nets. Main results are concisely presented in a table and our contribution is highlighted. We conclude with the analysis of an example drawn from the telecommunication industry underlining the efficiency brought by using minimal semiflows of minimal supports.