In Kunen's Set Theory, he gives a definition of $R_{\mathcal{I}}$ where $\mathcal{I}$ is an ideal.
If $R$ is a relation on $B$ and $\mathcal{I}$ is an ideal with dual filter $\mathcal{F}$ on $A$ and $f,g\in B^A$, then $f R_{\mathcal{I}}g$ iff $f R_{\mathcal{F}}g$ iff $\{a\in A: f(a) R f(b)\} \in \mathcal{F}.$
How do I interpret this definition? The third expression in the equivalence chain contains $b$ free, but the notation $f R_{\mathcal{I}}g$ does not have a $b$. Furthermore, the third expression does not contain $g$ at all. What is the intuition behind this?
Without more context I can’t be positive, but I’m reasonably sure that the last expression should be $\{a\in A:f(a)\mathrel{R}g(a)\}\in\mathscr{F}$. Intuitively, $f\mathrel{R_\mathscr{I}}g$ iff $f(a)$ and $g(a)$ are related for a ‘large’ set of $a\in A$. Here members of $\mathscr{I}$ are to be thought of as small subsets of $A$ and their complements as large subsets.