This chapter deals with the structure theory of Petri nets. The authors define the class of P/T systems, namely K-systems, for which the equivalence between controlled-siphon, deadlock-freeness, and liveness properties holds. Using the new structural notions of ordered transitions and root places, they revisit the non-liveness characterization of P/T systems satisfying the cs-property and define by syntactical manner new and more expressive subclasses of K-systems where the interplay between conflict and synchronization is relaxed.