EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓ_p AND ℓ_q

Abstract

We prove that, for 1 ≤ p < q < ∞, the partially ordered set P( )/Fin can be embedded into Borel equivalence relations between R / p and R / q . Since there is an antichain of size continuum in P( )/Fin, there are continuum many pairwise incomparable Borel equivalence relations between R / p and R / q .

Characterization of ‐like and c0‐like equivalence relations

“…[, Corollary 5.8]). Most recently, it was shown by Yin that itself can be embedded into .…”

On Schauder equivalence relations

“…Then the results were generalized by many writers in various ways by considering different background spaces such as Orlicz sequence spaces and (cf. ).…”

“…More recently, Yin [25] has moved further to embed whole P (ω)/Fin into the set of the equivalence relations between R N /l p and R N /l q to show the reducibility order of Borel equivalence relations between R N /l p and R N /l q are rather complex.…”

On Equivalence Relations Generated by Schauder Bases

“…Borel reducibility between equivalence relations of the form E f were investigated in [18]. Yin [22] generalized Mátrai's results to show that the partial order structure P (ω)/Fin can be embedded into the set of these E f 's equipped with the partial ordering of Borel reducibility. In fact, as noted in Yin [22], any such E f appeared in [22] is Borel bireducible to a Schauder equivalence relation R N /L, where L is an Orlicz sequence space.…”