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

Abstract

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