The Block Relation in Computable Linear Orders

TL;DR

This paper investigates the structure of computable linear orders, showing that certain dense orders without infinite ta-like intervals have computable copies with c.e. non-block relations, enabling non-trivial self-embeddings.

Contribution

It proves that all computable linear orders with dense condensation-type but no infinite ta-like intervals have computable copies with c.e. non-block relations, supporting a long-standing conjecture.

Findings

Every such linear order has a computable copy with a c.e. non-block relation.

Every computable linear order has a computable copy with a non-trivial self-embedding.

Supports the conjecture linking ta-like intervals to the existence of non-trivial self-embeddings.

Abstract

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