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It's known that the Dedekind–Macneille completion of an ordered Abelian group necessarily is not an ordered Abelian group (and it is an ordered Abelian monoid). I want to know that what happened whenever an ordered Abelian monoid is completed (by means od Dedekind–Macneille completion). Is the Dedekind–Macneille completion of an ordered Abelian monoid remains an ordered Abelian monoid?

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  • $\begingroup$ take the additive group of real numbers: what is the sum of $-\infty$ and $+\infty$, both in the completion? A better case is the positive cone of a directed abelian group. The concepts of "archimedean" and "integrally closed" are useful, compare Birkhoff, lattice theory. $\endgroup$ – user46855 Feb 15 '14 at 15:29
  • $\begingroup$ If you use instead the classical Dedekind completion (nonempty cuts), even for nonarchimedean totally ordered groups the monoid operation in the completion is non unique, see sciencedirect.com/science/article/pii/S016800720800078X For completions of monoids see eretrandre.org/rb/files/Erne1987_36.pdf $\endgroup$ – user46855 Feb 16 '14 at 22:16

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