Dichotomy theorems for families of non-cofinal essential complexity

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• 作者：John D. Clemens^a ; ^{johnclemens@boisestate.edu" class="auth_mail" title="E-mail the corresponding author} ; Dominique Lecomte^b ; ^c ; ^{dominique.lecomte@upmc.fr" class="auth_mail" title="E-mail the corresponding author} ; ; Benjamin D. Miller^d ; ^{benjamin.miller@univie.ac.at" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：primary ; 03E15 ; secondary ; 28A05
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：2 January 2017
• 年：2017
• 卷：304
• 期：Complete
• 页码：285-299
• 全文大小：378 K

文摘

We prove that for every Borel equivalence relation E, either E   is Borel reducible to E₀${\mathbb{E}}_0$, or the family of Borel equivalence relations incompatible with E has cofinal essential complexity. It follows that if F   is a Borel equivalence relation and F$\mathcal{F}$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation E  , either E∈F$E\in \mathcal{F}$ or F is Borel reducible to E  , then F$\mathcal{F}$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.