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In what follows, $\|\Phi\|_I$ means the Galvin-Hajnal rank of a function $\Phi:\kappa \to Ord$ with respect to a $\omega_1$-complete ideal $I$ of $\kappa$.

Lemma 2.2.5 of Introduction to Cardinal Arithmetic by Holz, Steffens and Weitz says:

Assume that $\kappa > \omega$ is a regular cardinal, $\Phi \in \,^\kappa Ord$, and $I$ is a $\kappa$-complete normal ideal on $\kappa$ with $\bigcup I=\kappa$. Then $\|\Phi\|_I \leq \sigma \Leftrightarrow \{\xi < \kappa : \Phi(\xi) \leq \sigma\} \notin I$ for every ordinal $\sigma < \kappa$.

The assumption on $\kappa$-completeness cannot be dropped. But can you drop one or more of the following assumptions:

  1. $\kappa$ is regular,
  2. $\bigcup I=\kappa$ and
  3. $I$ is normal?
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    @LostInMath, yes, in general if $\kappa \supsetneq \bigcup I = X \in I$, then f\ <_I\ g iff $g$ dominates $f$ on all of $\kappa - X$ and $|f|_I$ is the minimum value $f$ attains on $\kappa - X$.2011-06-18

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