Eventual domination on a regular cardinal ($f{<}^{\ast }g$)

For $f,g:\kappa \to \kappa$, write $f{<}^{\ast }g$ when there is $\gamma <\kappa$ such that $f(\delta )<g(\delta )$ whenever $\gamma <\delta <\kappa$. For an infinite regular cardinal, suprema of fewer than $\kappa$ ordinals below $\kappa$ remain below it, permitting the long chain under eventual domination construction.

For every infinite regular cardinal $\kappa$, there is a strictly eventually increasing sequence of ${\kappa }^{+}$ functions $\kappa \to \kappa$. At stage $0<\zeta <{\kappa }^{+}$ enumerate predecessors by $e:\kappa \to \zeta$ and put $f_{\zeta }(\delta )={\sup }_{\eta <\delta }(f_{e(\eta )}(\delta )+1)$. Regularity bounds each value below $\kappa$, while any predecessor's index is eventually included in the supremum.