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The Block Relation in Computable Linear Orders
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