I recently isolated the following definition, which I believe it should have appeared somewhere.
Let $\kappa$ be a cardinal, and let $X$ be a set with $\kappa^+\leq |X|$.
Definition: An ideal $\mathcal I\subseteq \mathcal P(\mathcal P_{\kappa^+}(X))$ is called a B-ideal if the following hold.
- for every $x\in X$, $\{A\in \mathcal P_{\kappa^+}(X):x\in A \}$ is not in $\mathcal I$.
- $(\mathcal I^{+},\subseteq)$ has a $\kappa$-closed dense subset which means there is $\mathcal D\subseteq \mathcal I^+$ such that for every $A\in\mathcal I^+$, there is $B\in\mathcal D$ with $B\subseteq A$, and that every decreasing sequence in $\mathcal D$ of length less than $\kappa$ has a lower bound.
In my definition B stands for Baumgartner.
My question is if such an ideal has a name in the litrature?
