刚性许多一度包含无限个1度的反链
Rigid many-one degrees contain infinite antichains of $1$-degrees
AI总结：
本文证明了m-刚性的集合其许多一度包含无限个1度的反链，并展示了该结论在随机实数中的应用。
摘要：
Odifreddi曾问是否每个非不可约的许多一度必须包含无限个一一度的反链。已知对于可计算枚举的许多一度（Degtev）和更近期的许多一度允许Δ^0_2代表（Batyrshin）有肯定答案。在本文中，我们隔离了这些现象背后的一种刚性原理。称一个集合A⊆ω为m-刚性，如果每个总可计算的m-自缩减都是最终恒等。我们证明如果A是m-刚性的，则其许多一度°_m(A)包含无限个1度的反链。证明使用了统一的复制构造：对于每个可计算参数S，我们定义B_S≡_m A，使得任何注入性缩减B_S≤_1 B_T诱导A的m-自缩减，并因此迫使S⊆*T。选择一个几乎包含无限个可计算集合的反链，从而在°_m(A)中得到所需的无限1度反链。作为应用，Jockusch的刚性定理表明每个1-通用集合是m-刚性的，从而给出一个稠密的家庭正例。此外，m-刚性在勒贝格测度下成立（实际上，每个马丁-洛夫随机实数都是m-刚性的）。因此，Odifreddi的问题5在公平硬币随机A∈2^ω的情况下有肯定答案；任何反例（如果存在）都被限制在零测度集（并通过通用性，也限制在稀疏集内）。
英文摘要：
Odifreddi asked whether every non-irreducible many-one degree must contain an infinite antichain of one-one degrees. Positive answers are known for computably enumerable many-one degrees (Degtev) and, more recently, for many-one degrees admitting a $Δ^0_2$ representative (Batyrshin). In this note we isolate a rigidity principle behind these phenomena. Call a set $A\subseteqω$ \emph{$m$-rigid} if every total computable $m$-autoreduction of $A$ is eventually the identity. We prove that if $A$ is $m$-rigid, then its many-one degree $°_m(A)$ contains an infinite antichain of $1$-degrees. The proof uses a uniform duplication construction: for each computable parameter $S$ we define $B_S\equiv_m A$ so that any injective reduction $B_S\le_1 B_T$ induces an $m$-autoreduction of $A$ and therefore forces $S\subseteq^{*}T$. Choosing an almost-inclusion infinite antichain of computable sets yields the desired infinite $1$-antichain inside $°_m(A)$. As applications, Jockusch's rigidity theorem implies that every $1$-generic set is $m$-rigid, giving a comeager family of positive instances. Moreover, $m$-rigidity holds with Lebesgue measure $1$ (indeed, every Martin-Löf random real is $m$-rigid). Consequently, Odifreddi's Question~5 has a positive answer \emph{with probability $1$} for a fair-coin random $A\in 2^ω$; any counterexample (if it exists) is confined to a null set (and, by genericity, also to a meager set).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
