A relative of the approachability ideal, diamond and non-saturation

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that □⁺ _λ together with 2^λ = λ⁺ implies Os for every S⊆λ⁺ that reflects stationarily often. In this paper, for a set S ⊆ λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. if I[S; λ] is fat, then NSλ⁺f S is non-saturated; 2. if I [S: λ] is fat and 2^λ = λ⁺, then ◇_s holds; 3. □^* _λ implies that I[S; λ] is fat for every S⊆λ⁺ that reflects stationarily often; 4. it is relatively consistent with the existence of a supercompact cardinal that □^z.ast; _λ fails, while I[S; λ] is fat for every stationary S⊆λ⁺ that reflects stationarily often. The stronger principle ◇ ^* λ⁺is studied as well.