Infima and cardinal characteristics of critical ideals for countable compact spaces

Abstract: For each countable ordinal $α\ge 2$, the ideals $\mathsf{conv}α$ were introduced in ``Critical ideals for countable compact spaces'' (to appear in Fund. Math., see also arXiv:2503.12571) to characterize compact countable spaces homeomorphic to $ω^α\cdot n+1$ with the order topology. We study the structure of these ideals in the Katětov order, namely for limit ordinals $α$, we show that $\mathsf{conv}α$ do not serve as greatest lower bounds of the $\mathsf{conv}β$ for $β<α$. We therefore define the ideals $\mathsf{conv}{<α}$ with this property and show that together, the ideals $\mathsf{conv}α$ and $\mathsf{conv}{<α}$ form intertwined decreasing hierarchies of $Σ^0_4$- and $Π^0_5$-complete ideals. Furthermore, we examine several cardinal invariants of $\mathsf{conv}_α$, computing invariants that have recently appeared in the literature in various contexts.