The Block Relation in Computable Linear Orders

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 non-block relation ¬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 η-like interval) holds for all linear orders with dense condensation-type.