Notre Dame Journal of Formal Logic

The Block Relation in Computable Linear Orders

Michael Moses

Abstract

The block relation B(x,y) in a linear order is satisfied by elements that are finitely far apart; a block is an equivalence class under this relation. We show that every computable linear order with dense condensation-type (i.e., a dense collection of blocks) but no infinite, strongly η-like interval (i.e., with all blocks of size less than some fixed, finite k) has a computable copy with the nonblock relation ¬B(x,y) computably enumerable. This implies that every computable linear order has a computable copy with a computable nontrivial self-embedding and that the long-standing conjecture characterizing those computable linear orders every computable copy of which has a computable nontrivial self-embedding (as precisely those that contain an infinite, strongly η-like interval) holds for all linear orders with dense condensation-type.

Article information

Source
Notre Dame J. Formal Logic, Volume 52, Number 3 (2011), 289-305.

Dates
First available in Project Euclid: 28 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1311875775

Digital Object Identifier
doi:10.1215/00294527-1435465

Mathematical Reviews number (MathSciNet)
MR2822490

Zentralblatt MATH identifier
1260.03085

Subjects
Primary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]
Secondary: 03C57: Effective and recursion-theoretic model theory [See also 03D45]

Keywords
computable linear order block relation self-embedding

Citation

Moses, Michael. The Block Relation in Computable Linear Orders. Notre Dame J. Formal Logic 52 (2011), no. 3, 289--305. doi:10.1215/00294527-1435465. https://projecteuclid.org/euclid.ndjfl/1311875775


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