Abstract non-piecewise syndetic sets

We introduce an abstract scheme (Fi)i∈I which can be considered as a framework for defining all possible versions of ideals of non-piecewise syndetic sets in a common language. We prove that the family ∇((iFi∈I)) = {Z:i∈I ∀j∈I ∃F∈Fj ∀G∈Fi ∃ G ⊆ F ∧ G ∩ Z = ∅} always forms an ideal. It is shown that, under a certain additional condition, this ideal is equal to the collection of non-piecewise syndetic sets relative to (iFi∈I) . This correspondence holds in many important cases, including the integer lattice. Moreover, we prove that anyσ F ideal has such representation with finite sets. Finally, we show that the Marczewski-Burstin representation is a particular case of our general abstract scheme.