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

Patrizio Cintioli

Abstract

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 \(ω\)-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.

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

Abstract

We construct a nonrecursive set and a uniformly computable family of sets , all bounded finite-one equivalent to , such that the corresponding -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 -chain and asks whether there are bounded finite-one degrees consisting exactly of a dense linearly ordered set of -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 -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 -rigid, so the phenomena obtained in this paper arise from a mechanism different from earlier -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.

Paper Structure

This paper contains 5 sections, 6 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. A Profile Theorem for Block-Density Codings
  4. Universal order embeddings
  5. Concluding Remarks

Key Result

Theorem 3.2

Let $\{u_i\}_{i\in\mathbb N}$ be a uniformly computable family of block-density profiles, and let $D\subseteq\mathbb N^2$ be a computable set of ordered pairs. Assume that for each $(i,j)\in D$ we are given, uniformly computably, such that for every $k\in S_{i,j}$. Then there exist a set $A\subseteq\mathbb N$, computable in $\emptyset'$, and a uniformly computable family of total surjective func

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2: Profile Theorem
  • proof
  • Remark 3.3
  • Corollary 3.4: Constant profiles
  • proof
  • Corollary 3.5: Oscillating profiles
  • proof
  • Corollary 3.6: One degree containing both structures
  • ...and 8 more