## The block relation in computable linear orders.(English)Zbl 1260.03085

Summary: 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 $$\eta$$-like interval (i.e., with all blocks of size less than some fixed, finite $$k$$) has a computable copy with the nonblock relation $$\lnot B(x,y)$$ 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 $$\eta$$-like interval) holds for all linear orders with dense condensation-type.

### MSC:

 03D45 Theory of numerations, effectively presented structures 03C57 Computable structure theory, computable model theory

### Keywords:

computable linear order; block relation; self-embedding
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