##
**Cofinal families of Borel equivalence relations and quasiorders.**
*(English)*
Zbl 1102.03045

The binary relation \(R\) on the Polish space \(X\) Borel reduces to the binary relation \(S\) on the Polish space \(Y\), written \(R\leq_ B S\), if there is a Borel function \(f: X\rightarrow Y\) such that \(xRy\) if and only if \(f(x)Sf(y)\). The case where \(R\) and \(S\) are both equivalence relations is often studied. In this paper the author also allows \(R\) and \(S\) to be quasiorders (reflexive and transitive binary relations).

The author first shows that there is an analytic ideal on \(\omega\) whose induced equivalence relation on \(2^\omega\) is complete analytic. As a consequence, any Borel equivalence relation reduces to one induced by a Borel ideal. This answers a question of Kanovei.

The author goes on to study operations on equivalence relations and quasiorders that in some instances lead to jumps in the Borel hierarchy. He then shows that a certain jump induces cofinal families of Borel equivalence relations and quasiorders with respect to \(\leq_ B\). This is done by generalizing Scott’s analysis of isomorphisms to quasiorders. Finally, several Borel equivalence relations, including the Lipsschitz isomorphism of compact metric spaces, are shown to be \(K_\sigma\) complete.

The author first shows that there is an analytic ideal on \(\omega\) whose induced equivalence relation on \(2^\omega\) is complete analytic. As a consequence, any Borel equivalence relation reduces to one induced by a Borel ideal. This answers a question of Kanovei.

The author goes on to study operations on equivalence relations and quasiorders that in some instances lead to jumps in the Borel hierarchy. He then shows that a certain jump induces cofinal families of Borel equivalence relations and quasiorders with respect to \(\leq_ B\). This is done by generalizing Scott’s analysis of isomorphisms to quasiorders. Finally, several Borel equivalence relations, including the Lipsschitz isomorphism of compact metric spaces, are shown to be \(K_\sigma\) complete.

Reviewer: J. M. Plotkin (East Lansing)

### MSC:

03E15 | Descriptive set theory |

### References:

[1] | DOI: 10.2307/2687754 · Zbl 0994.54037 · doi:10.2307/2687754 |

[2] | DOI: 10.4064/fm172-2-3 · Zbl 1029.46009 · doi:10.4064/fm172-2-3 |

[3] | Back and forth through infinitary logic (1973) |

[4] | Harvey Friedman’s research on the foundations of mathematics pp 11– (1985) |

[5] | Complete analytic equivalence relations 357 pp 4839– (2005) · Zbl 1118.03043 |

[6] | Comptes Rendus de l’Académie des Sciences. Série I 333 pp 903– (2001) |

[7] | Fundamenta Mathematicae 164 pp 61– (2000) · Zbl 0958.68177 |

[8] | Classical descriptive set theory (1995) |

[9] | Izvestiya: Mathematics 67 (2003) |

[10] | Analytic equivalence relations and Ulm-type classifications 60 pp 1273– (1995) |

[11] | On the classification of Polish metric spaces up to isometry 161 (2003) · Zbl 1012.54038 |

[12] | A Borel reducibility theory for classes of countable structures 54 pp 894– (1989) · Zbl 0692.03022 |

[13] | DOI: 10.1090/S0894-0347-97-00221-X · Zbl 0865.03039 · doi:10.1090/S0894-0347-97-00221-X |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.