Dense Chains, Antichains, and Universal Partial Orders Inside a Bounded Finite-One Degree
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915898204356608 |
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| author | Cintioli, Patrizio |
| author_facet | Cintioli, Patrizio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27901 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dense Chains, Antichains, and Universal Partial Orders Inside a Bounded Finite-One Degree Cintioli, Patrizio Logic 03D30, 06A06 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. |
| title | Dense Chains, Antichains, and Universal Partial Orders Inside a Bounded Finite-One Degree |
| topic | Logic 03D30, 06A06 |
| url | https://arxiv.org/abs/2603.27901 |