We solve a longstanding question of Rosenstein, and make progress toward solving a long-standing open problem in the area of computable linear orderings by showing that every computable η-like linear ordering without an infinite strongly η-like interval has a computable copy without nontrivial computable self-embedding. The precise characterization of those computable linear orderings which have computable copies without nontrivial computable self-embedding remains open.
Rodney G. Downey. Bart Kastermans. Steffen Lempp. "On computable self-embeddings of computable linear orderings." J. Symbolic Logic 74 (4) 1352 - 1366, December 2009. https://doi.org/10.2178/jsl/1254748695