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I'm looking for a reference that surveys (as generally as possible) the various notions of 'completion' for an ordered space and their relation to eachother.

In particular, I'd like a reference that covers notions like pseudo-convergent completion in relation to notions like Dedekind and Archimedean completion for ordered fields. I'm aware of the excellent paper by Philip Ehrlich discussing the latter two notions and their relationship in depth, but it does not appear to touch on pseudo-convergent completion or its relation to the other two.

Even more particularly, I'd like a reference on how the value class of a non-Archimedean field is effected by the various notions of completion. Pseudo-convergent completion leaves the value class unchanged while possibly adding new elements with a given valuation to the underlying field; I would like to know if other types of completion can canonically extend the value class of a non-Archimedean ordered field.

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For Dedekendian completions and (transfinite) Cantor completions and the fact that they preserve the value group, see the following classical works of COHEN and GOFFMAN.

COHEN, L. W. and GOFFMAN, C., A theory of transfinite convergence, Trans. Amer. Math. Soc. 66 (1949), 65–74.

COHEN, L. W. and GOFFMAN, C., The topology of ordered Abelian groups, Trans. Amer. Math. Soc. 67 (1949), 310–319.

COHEN, L. W. and GOFFMAN, C., On completeness in the sense of Archimedes, Amer. J. Math. 72 (1950), 747–751.

For other excellent references that might be of aid to you, see:

HOLLAND, C., Extensions of ordered groups and sequence completions, Trans. Am. Math. Soc. 107 (1963), 71–82.

KIJIMA D. and NISHI, M., The pseudo-convergent sets and the cuts of an ordered field, Hiroshima Math. J. 19 (1989), 89–98.

F.-V. KUHLMANN, SELECTED METHODS FOR THE CLASSIFICATION OF CUTS, AND THEIR APPLICATIONS, arXiv:1803.07806v1 [math.AC] 21 Mar 2018

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  • $\begingroup$ Are you aware of any kinds of completion besides algebraic completion that do extend the value class of a field? $\endgroup$ – Alec Rhea May 20 '18 at 6:34
  • $\begingroup$ I meant algebraic closure* but can’t edit now. $\endgroup$ – Alec Rhea May 20 '18 at 6:40

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