It is consistent that there is a partial order (P,≤) of size such that every monotone function f:P → P is first order definable in (P,≤).
@article{bwmeta1.element.bwnjournal-article-fmv152i3p255bwm, author = {Martin Goldstern and Saharon Shelah}, title = {A partial order where all monotone maps are definable}, journal = {Fundamenta Mathematicae}, volume = {154}, year = {1997}, pages = {255-265}, zbl = {0876.03027}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv152i3p255bwm} }
Goldstern, Martin; Shelah, Saharon. A partial order where all monotone maps are definable. Fundamenta Mathematicae, Tome 154 (1997) pp. 255-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv152i3p255bwm/
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