On Smital properties

Let A be an algebra and J ⊂ A an ideal of subsets of a group 〈 X , + 〉 with an invariant topology τ. We say that a triple 〈 A , J , τ 〉 has the Smital property if, for any set A ∈ A ∖ J and any set D dense in τ, the set ( A + D ) c belongs to J . In the paper we compare this property and similar ones with the well-known Steinhaus type properties. We consider several weak and strong versions of Smital properties.