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

TL;DR

This paper constructs a bounded finite-one degree with complex order-theoretic structures, including dense linear orders, antichains, and embeddings of all countable partial orders, using a novel block-density profile method.

Contribution

Introduces a new block-density profile method to control one-one reducibility within a bounded finite-one degree, enabling complex order embeddings.

Findings

Constructed a bounded finite-one degree containing a dense linear order.

Embedded an infinite antichain of 1-degrees within a single degree.

Demonstrated that such degrees can embed all countable partial orders.

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 \(\omega\)-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.