arXiv subjects

Patrizio Cintioli

Publications and source records attributed to Patrizio Cintioli.

9 recordsLinked to original sources

arXiv2026

A Bounded Finite-One Degree Whose One-One Degrees Form Exactly a Dense Linear Order

We construct a set $U\leq_T\emptyset''$ whose bounded finite-one degree, ordered internally by one-one reducibility, consists exactly of a countable dense linear order without endpoints. More precisely, \[ \left( \{[B]_1:B\equiv_{\mathrm{bfo}}U\},\leq_1 \right) \cong (\mathbb Q,\leq). \] This exact realization contrasts with two previous results. In earlier work, a bounded finite-one degree was constructed that contains a copy of $(\mathbb Q,\leq)$, but the same degree also contains an infinite antichain of one-one degrees and, more generally, embedded copies of all countable partial orders; thus the dense chain does not exhaust the degree. In a different direction, $m$-rigidity yields an almost-sure and comeager obstruction: for a measure-$1$ and comeager class of sets, the corresponding bounded finite-one degree contains an infinite antichain of one-one degrees and hence is not linearly ordered. The present construction shows that, despite this typical negative behaviour, exact dense linear order can occur. In particular, it gives an affirmative answer to Open Question~3 of Richter, Stephan, and Zhang. The proof has two main parts. A block homogenization construction produces a noncylindrical set $U$, a weak dyadic tower $(U_e)_{e\in\mathbb N}$, computable reservoirs of both colours, and a base absorption property. An abstract absorption-to-exhaustivity theorem then shows that every member of the bounded finite-one degree of $U$ is one-one equivalent to some finite autojoin $mU_e$, thereby yielding the exhaustive classification above.

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arXiv2026

Every Nonrecursive Many-One Degree Contains Either One or Infinitely Many Finite-One Degrees

This paper proves that every nonrecursive many-one degree contains either exactly one or infinitely many finite-one degrees. This gives a negative answer to Open Question 2 of Richter, Stephan, and Zhang~\cite[p.~17]{RSZ}. Earlier work established that, for almost every $A\subseteq\N$ with respect to the standard product measure on $2^\N$, the degree $[A]_{\m}$ contains an infinite antichain of finite-one degrees~\cite{Cintioli}. Thus the question had already received an almost-sure negative answer; the present theorem settles it for every nonrecursive many-one degree.

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arXiv2026

A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees

In this paper, we provide a negative solution to Problem 3 formulated by P.~Odifreddi in his survey articles \textit{``Strong Reducibilities''} (1981) and \textit{``Reducibilities''} (1999). The problem asks whether every computably enumerable (c.e.) $tt$-degree contains a c.e.\ \textit{irreducible} $m$-degree (i.e., an $m$-degree consisting of only one $1$-degree). We answer this question in the negative by proving the existence of a c.e.\ $tt$-degree that does not contain any c.e.\ irreducible $m$-degree. Our proof relies on the structural properties of c.e.\ semirecursive sets with a rigid complement, originally constructed by A.~N.~Degtev. We show that the unique c.e.\ $m$-degree contained within the $tt$-degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple $1$-degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. -- which guarantees the existence of an irreducible $m$-degree within every c.e.\ $tt$-degree -- is strictly optimal and cannot be generally strengthened to require such an $m$-degree to be computably enumerable.

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arXiv2026

A strengthened form of D\"egtev's argument and $D$-maximal many-one degrees

D\"egtev proved that if a noncomputable computably enumerable set $U$ satisfies his property $(R)$ and the many-one degree of $U$ contains no simple set, then the $1$-degree of $U$ is minimal. We show that his proof yields the stronger conclusion that, for every noncomputable $A\lem U$, \[ U\le_1 A. \] In particular, the injectivity of the reduction $A\le_1 U$ with which D\"egtev begins his proof is not needed. Using the Cholak--Gerdes--Lange classification, we show that for a noncomputable $D$-maximal set $D$, the hypothesis that its many-one degree contains no simple set is equivalent to $D$ being of one of Types~3--10. Hence, for every such $D$ and every noncomputable $A\lem D$, one has $D\le_1 A$. Together with Maslova's theorem for the simple-set cases, this recovers as a corollary the existence of a least finite-one degree in every noncomputable c.e.\ many-one degree containing a $D$-maximal set.

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arXiv2026

A computably enumerable many-one degree with no least finite-one degree

Richter, Stephan, and Zhang asked whether every nonrecursive many-one degree contains a least finite-one degree. We solve this question in the negative, already within the class of computably enumerable many-one degrees. Positive answers are known in two disjoint natural settings: for a measure-one and comeager class of $m$-rigid sets, and, in a companion paper, for computably enumerable many-one degrees containing a $D$-maximal set. We construct a nonrecursive \ce\ set $A$ such that for every set $X \eqm A$ there exists a c.e.\ set $B \eqm A$ with $X \not\lfo B$. Hence the many-one degree of $A$ contains no least finite-one degree. The proof is a finite-injury priority construction based on virtual target sets and a dynamic trap mechanism forcing any putative finite-one reduction either to violate finite-oneness or to compute an incorrect reduction.

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arXiv2026

Dense Chains, Antichains, and Universal Partial Orders Inside a Bounded Finite-One Degree

We construct a nonrecursive set $A\le_T\emptyset'$ and a uniformly computable family of sets $C_0,C_1,\dots$, all bounded finite-one equivalent to $A$, such that the corresponding $1$-degrees form a copy of the dense linear order $(\mathbb Q,\le)$. Motivated by a recent preprint of Richter, Stephan, and Zhang, which shows that bounded finite-one degrees can be as rigid as a discrete $\omega$-chain and asks whether there are bounded finite-one degrees consisting exactly of a dense linearly ordered set of $1$-degrees, we introduce a block-density profile method for controlling one-one reducibility inside a single bounded finite-one degree. As further applications, in the same bounded finite-one degree we obtain an infinite antichain of $1$-degrees and, more generally, an embedded copy of every countable partial order. A single bounded finite-one degree can already exhibit dense, incomparable, and universal order-theoretic behaviour. Our main technical tool is a profile theorem based on computable block-density codings. The witness set constructed here is not $m$-rigid, so the phenomena obtained in this paper arise from a mechanism different from earlier $m$-rigidity-based constructions. Although our results do not settle the exact realization problem posed by Richter, Stephan, and Zhang, we show that density itself is not the obstruction: a single bounded finite-one degree may already contain a copy of $(\mathbb Q,\le)$, an infinite antichain, and embeddings of all countable partial orders.

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arXiv2026

A $wtt$-introimmune set in \texorpdfstring{$\Pi^0_1$}{Pi01} and introimmunity for several reducibilities

We prove that there exists a weak truth-table introimmune set in the class $\Pi^0_1$, settling the question left open in previous work of whether the known $\Delta^0_2$ existence result can be improved to $\Pi^0_1$. Since $\Sigma^0_1$ sets cannot be immune, this is best possible for weak truth-table introimmunity. We also study introimmunity for Jockusch's bounded-search reducibility $\le_{bs}$ and Andersen's Dartmouth reducibility $\le_D$, proving the existence of $\Delta^0_2$ sets that are $bs$-introimmune and $D$-introimmune; hence there also exists a $\Delta^0_2$ $D^+$-introimmune set. We next consider the classical reducibility $\le_Q$, which is not contained in $\le_T$ on all subsets of $\omega$. We show that no infinite $\Pi^0_1$ set is $Q$-introimmune, while a $\Delta^0_2$ $Q$-introimmune set does exist. Thus the existence of $\Delta^0_2$ $Q$-introimmune sets is best possible within the arithmetical hierarchy. Finally, for enumeration reducibility $\le_e$, we show that no infinite $\Pi^1_1$ set is $e$-introimmune, although $e$-introimmune sets do exist in the unrestricted sense. The proofs combine finite-injury priority arguments with dynamic spacing methods for $\le_{wtt}$, $\le_{bs}$, and $\le_D$, a bit-by-bit finite-extension construction for $\le_Q$, and an application of Soare's abstract existence theorem in the enumeration case.

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arXiv2026

$m$-Rigidity and Finite-One Degrees Inside Typical Many-One Degrees

In recent work, the notion of $m$-rigidity was introduced as a sufficient condition for the existence of infinite antichains of $1$-degrees inside many-one degrees. Motivated by a recent preprint of Richter, Stephan, and Zhang on finite-one degrees inside many-one degrees, we study the finite-one structure of the many-one degree of an $m$-rigid set. First, combining bi-immunity of $m$-rigid sets with a theorem of Richter, Stephan, and Zhang, we show that for Lebesgue-almost every set $A$, and for a comeager class of sets $A$, the many-one degree $\deg_m(A)$ contains a least finite-one degree. Second, we prove that if $A$ is $m$-rigid, then $\deg_m(A)$ contains infinitely many pairwise incomparable finite-one degrees. More precisely, we construct representatives $B_S \equiv_m A$, indexed by computable sets $S$, such that $T \setminus S$ infinite implies $B_T \not\leq_{fin} B_S$. Third, inside a single finite-one degree we build a strict ascending chain \[ A_{(1)} <_1 A_{(2)} <_1 \cdots \] of $1$-degrees. These results yield almost-sure and comeager partial answers to the first two open problems posed by Richter, Stephan, and Zhang.

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arXiv2026

Rigid many-one degrees contain infinite antichains of $1$-degrees

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 $\Delta^0_2$ representative (Batyrshin). In this note we isolate a rigidity principle behind these phenomena. Call a set $A\subseteq\omega$ \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 $\deg_m(A)$ 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 $\deg_m(A)$. 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-L\"of random real is $m$-rigid). Consequently, Odifreddi's Question~5 has a positive answer \emph{with probability $1$} for a fair-coin random $A\in 2^\omega$; any counterexample (if it exists) is confined to a null set (and, by genericity, also to a meager set).

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