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/
[00000] [KS] H. Kaiser and N. Sauer, Order polynomially complete lattices, Algebra Universalis 130 (1993), 171-176. | Zbl 0784.06004
[00001] [Sh 128] S. Shelah, Uncountable constructions for B.A., e.c. groups and Banach spaces, Israel J. Math. 51 (1985), 273-297.
[00002] [Sh 136] S. Shelah, Constructions of many complicated uncountable structures and Boolean algebras, Israel J. Math. 45 (1983), 100-146.