  Using the notions of Topological dynamics, H. Furstenberg defined central
sets and proved the Central Sets Theorem. Later V. Bergelson and N. Hindman
characterized central sets in terms of algebra of the Stone-\v{C}ech
compactification of discrete semigroup. They found that central sets are the
members of the minimal idempotents of \b{eta}S, the Stone-\v{C}ech
compactification of a semigroup (S, .). Hindman and leader introduced the
notion of Central set near zero algebraically. Later dynamical and
combinatorial characterization have also been established. For any given filter
F in S a set A is said to be a F- central set if it is a member of a minimal
idempotent of a closed subsemigroup of \b{eta}S, generated by the filter F. In
a recent article Bergelson, Hindman and Strauss introduced strongly central and
very strongly central sets in [BHS]. They also dynamically characterized the
sets in the same paper. In the present article we will characterize the
strongly F- central sets dynamically and combinatorially. Here we introduce the
filter version of strongly central sets and very strongly central sets. We also
provide dynamical and combinatorial characterization of such sets.
