On the Presburger fragment of logics with multiteam semantics

In team semantics, which is the basis of modern logics of dependence and independence, formulae are evaluated on sets of assignments, called teams. Multiteam semantics instead takes mulitplicities of data into account and is based on multisets of assignments, called multiteams. Logics with multiteam semantics can be embedded into a two-sorted variant of existential second-order logics, with arithmetic operations on multiplicities. Here we study the Presburger fragment of such logics, permitting only addition, but not multiplication on multiplicities. It can be shown that this fragment corresponds to inclusion-exclusion logic in multiteam semantics, but, in contrast to the situation in team semantics, that it is strictly contained in independence logic. We give different characterisations of this fragment by various atomic dependency notions.