We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.

@article{paperbot1290,
  title = {On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics},
  author = {Matthias Hoelzel and Richard Wilke},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 17, Issue 3},
  year = {2021},
  doi = {10.46298/lmcs-17(3:14)2021}
}