Reducibility of Equivalence Relations Arising from Nonstationary Ideals under Large Cardinal Assumptions

Abstract

Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal $\kappa$. We show the consistency of $E_{\lambda \text{-}\mathrm{club}}^{{\lambda }^{++},{\lambda }^{++}}$, the relation of equivalence modulo the nonstationary ideal restricted to $S_{\lambda }^{{\lambda }^{++}}$ in the space $({\lambda }^{++}{)}^{{\lambda }^{++}}$, being continuously reducible to $E_{{\lambda }^{+}\text{-}\mathrm{club}}^{2,{\lambda }^{++}}$, the relation of equivalence modulo the nonstationary ideal restricted to $S_{{\lambda }^{+}}^{{\lambda }^{++}}$ in the space $2^{{\lambda }^{++}}$. Then we show that for $\kappa$ ineffable $E_{reg}^{2,\kappa }$, the relation of equivalence modulo the nonstationary ideal restricted to regular cardinals in the space $2^{\kappa }$ is ${\Sigma }_1^1$-complete. We finish by showing that, for ${\Pi }_2^1$-indescribable $\kappa$, the isomorphism relation between dense linear orders of cardinality $\kappa$ is ${\Sigma }_1^1$-complete.