  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.
