Ideals with Smital properties

Abstract A $$\sigma $$ σ -ideal $$\mathcal {I}$$ I on a Polish group $$(X,+)$$ ( X , + ) has the Smital Property if for every dense set D and Borel -positive B algebraic sum $$D+B$$ D B is complement of from . We consider several variants this property study their connections with countable chain condition, maximality how well they are preserved via Fubini products. In particular we show that there $$\mathfrak {c}$$ c many maximal invariant -ideals bases Cantor space $$2^\omega 2 ω