Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that together with 2λ = λ+ implies ⋄S 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 + ∣ S is non-saturated;
2. if I[S; λ] is fat and 2λ = λ+, then ⋄S holds;
3. 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 fails, while I[S; λ] is fat for every stationary S ⊆ λ+ that reflects stationarily often.
The stronger principle is studied as well.
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