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

Abstract

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that □^*_λ together with 2^λ=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 NS_λ⁺↾ 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 □^*_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.