Postprint version. Published in Topology and its Applications, Volume 47, Issue 1, November 19, 1992, pages 35-52.
The definitive version is available at https://doi.org/10.1016/0166-8641(92)90113-E.
We construct the Nachbin ordered compactification and the ordered realcompactification, a notion defined in the paper, of a given ordered topological space as nonstandard ordered hulls. The maximal ideals in the algebras of the differences of monotone continuous functions are completely described. We give also a characterization of the class of completely regular ordered spaces which are closed subspaces of products of copies of the ordered real line, answering a question of T.H. Choe and Y.H. Hong. The methods used are topological (standard) and nonstandard.