We show that a set has an η-representation in a linear order if and only if it is the range of a 0'-computable limitwise monotonic function. We also construct a Δ₃ Turing degree for which no set in that degree has a strong η-representation, answering a question posed by Downey.
"η-representation of sets and degrees." J. Symbolic Logic 73 (4) 1097 - 1121, December 2008. https://doi.org/10.2178/jsl/1230396908