Embeddings of $P(\omega)/{\rm Fin}$ into Borel Equivalence Relations between $\ell_p$ and $\ell_q$

Abstract

We prove that, for $1 \le p<q<\infty$, the partially ordered set $P(\omega)/{\rm Fin}$ can be embedded into Borel equivalence relations between $\mathbb{R}^\omega/\ell_p$ and $\mathbb{R}^\omega/\ell_q$. Since there is an antichain of size continuum in $P(\omega)/{\rm Fin}$, therefore there are continuum many incomparable Borel equivalence relations between $\mathbb{R}^\omega/\ell_p$ and $\mathbb{R}^\omega/\ell_q$.