Generating Sets of Order Preserving Mappings

Sierpiński proved in 1935 that any countable set of mappings on an infinite set X is contained in a subsemigroup of the semigroup of all maps on X, where this subsemigroup is generated by just two mappings. Since then, it has been of interest whether the result also holds when certain restrictions are placed on the properties of the maps. In this paper we investigate the minimum number of order preserving maps on the extended reals or rationals needed to generate any countable set of such mappings. We prove that five such maps are sufficient but two are not. It remains to be investigated whether any countable set of order preserving mappings on the extended reals or rationals could be generated by four or even three such maps. In the last section, we prove that not every countable set of order preserving maps on the naturals can be generated by a finite number of such maps.