Certain generating subspaces for vertex operator algebras

We study minimal generating subspaces of "weak PBW type" for vertex operator algebras that satisfy a certain condition on the grading, and we develop a procedure for finding such subspaces. As applications, we derive some results on generalized modules and produce a minimal generating space of weak PBW type for the vertex operator algebra $V_L$ associated with any positive-definite even lattice $L$.