  Moonshine relates three fundamental mathematical objects: the Monster
sporadic simple group, the modular function j, and the moonshine module vertex
operator algebra. Examining the relationship between modular functions and the
representation theory of vertex operator algebras reveals rich structure. In
particular, C2-cofiniteness (also called Zhu's finiteness condition) implies
the existence of finite generating sets and Poincare-Birkhoff-Witt-like
spanning sets for vertex operator algebras and their modules. These spanning
sets feature desirable ordering restrictions, e.g., a difference-one condition.
