Splitting positive sets

We introduce a class of cardinal invariants 𝔰_ I for ideals I on ω which arise naturally from the FinBW property introduced by Filipów et al. (2007). Let I be an ideal on ω . Define 𝔰_ I = min{ |X|:X ⊂[ω ]^ω,∀ B ∈ I^ + , ∃ x ∈ X(B\ x,B ∩ x ∈[ω ]^ω)}. We characterize them and compare them with other cardinal invariants of the continuum.