Abstract non piecewise syndetic sets

Abstract

We create an abstract schema $(\cF_{i})_{i\in I}$ which can help us to define all possible
versions of ideals of non-piecewise syndetic sets in a common language.
We prove that the family $\nabla((\cF_{i})_{i\in I}) = \lbrace Z \colon
\forall_{i\in I} \exists_{j\in I} \forall_{F \in \cF_j}
\exists_{G\in \cF_i} G \subseteq F \wedge G \cap Z = \emptyset\rbrace$
is always an ideal. We prove that under special condition this ideal
is equal to the non piecewise syndetic sets relative to $(\cF_{i})_{i\in I}$.
This holds in many special cases, in particular in the case of
the integer lattice. We prove that any $F_{\sigma}$ ideal has
such representation with finite sets. Finally we prove that
the Marczewski-Burstin representation is a particular case
of our general abstract scheme.