Embeddings Of P(ω)/Fin Into Equivalence Relations Between L_p And L_q

Borel reducibility is a fundamental concept in Descriptive Set Theory, we often use it to compare the complexity of different equivalence relations. In all these equivalence relations,lp-like(p≥1) equivalence relations play an important role. R. Dougherty and G. Hjorth [9] proves that, for1≤p≤q<∞, lp equivalence relation is Borel reducible to lp equivalence relation. This is a nice result, but we still want to know how complicated of the reductions between these lp-like(p≥1) equivalence relations, for example, S. Gao [13] asked whether for1≤p<∞, lp equivalence relation is the largest lower bound of{lp:p<q} equivalence relations. If the answer is negative, then for1≤p≤q<∞, how complicated is the Borel reduction structure between lp and lp?Let f:[0,1]â†'R+be an arbitrary function, we consider the relation Ef on [0,1]ω defined by setting, for every x, y∈[0,1]ω,(x,y)∈Ef (?)∑n<ω, f(|y(n)-x(n)|)<oo. When f is a Borel function and Ef is an equivalence relation, we give some results about reducibility and nonreducibility between Ef’s. Furthermore, we prove that the partial order structure (P(ω)/Fin,(?)) can be embedded into Ef’s. Using this result, we show that for all1≤p≤q<∞,(P(ω)/Fin,(?)) can be embedded into the equivalence relations between lp equivalence relation and lq equivalence relation, which implies that there is continuum many Borel incomparable equivalence relations between lp equivalence relation and lq equivalence relation.The structure of this dissertation is arranged as follows:The first chapter is the introduction. In this chapter, we give a brief introduction on the historical background of classical descriptive set theory, effective descriptive set theory and invariant descriptive set theory, then we give some basic definitions about equivalence relations and Borel reducibility. Finally we give the figure about reducibil- ity between some landmark equivalence relations, then introduce the background and main results of this dissertation briefly.The second chapter is the preliminaries. In this chapter, we give some basic notions that will be used in the following chapters. Simultaneously, we list some known results about reducibility and nonreducibility between Ef’s.The third chapter is the results about reducibility and nonreducibility between Ef’s. In this chapter, we mainly give some results about reducibility and nonreducibility between Ef’s which will be easily facilitated, these results are deducted from the known results in chapter2. At the same time, we discuss some properties about approximation of functions and sequences.The fourth chapter is the core part of this dissertation. In this chapter, we mainly prove two important results:one is that partial order structure (P(ω)/Fin,(?)) can be embedded into Ef’s, the other is that for all1≤p<q<∞,(P(ω)/Fin, C) can be embedded be embedded into the equivalence relations between lp equivalence relation and lq equivalence relation, which implies there is continuum many Borel incompara-ble equivalence relations between lp equivalence relation and lq equivalence relation. Finally, we give an independent proof for this.