In mathematics, Pataraia's theorem states each monotone map f : P → P {\displaystyle f:P\to P} for an inductive poset P {\displaystyle P} admits a least fixed point, where an inductive poset means a dcpo with a least element. It was introduced by Dito Pataraia in 1997. The theorem is a variant of the Bourbaki–Witt theorem.
See also Markowsky's theorem
References
Paul Taylor, A Fixed Point Theorem for Categories [2] Jean Goubault-Larrecq, Bourbaki, Witt, and Dito Pataraia, [3] Andrej Bauer, Peter LeFanu Lumsdaine, On the Bourbaki-Witt Principle in Toposes, Math. Proc. Cam. Phil. Soc. 155 (2013), no. 1, 87–99 doi:10.1017/S0305004113000108, arXiv:1201.0340. [4] Martin Escardo. Joins in the frame of nuclei. Applied Categorical Structures 11: 117–124, 2003. https://doi.org/10.1023/A:1023555514029 https://martinescardo.github.io/TypeTopology/Various.Pataraia.html
Further reading https://ncatlab.org/nlab/show/fixed+point#pataraias_theorem https://golem.ph.utexas.edu/category/2012/10/the_curious_dependence_of_set.html#c042391