View source for Dedekind completion

''of a Riesz space''

A [[Riesz space|Riesz space]] is called Dedekind complete if every non-empty subset that is bounded from below (respectively, above) has an infimum (respectively, supremum). A Dedekind-complete Riesz space is automatically Archimedean. Hence, so are its Riesz subspaces.

Given an Archimedean Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100501.png" />, a Dedekind completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100502.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100503.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100504.png" /> is a Riesz space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100505.png" /> is a mapping such that

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100506.png" /> is Dedekind complete;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100507.png" /> is a Riesz isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100508.png" /> onto a Riesz subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d1100509.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005010.png" />;

3) as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005012.png" /> is normal, i.e., it preserves arbitrary suprema and infima;

4) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005013.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005014.png" /></td> </tr></table>

Every Archimedean Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005015.png" /> has a Dedekind completion, whose underlying [[Partially ordered set|partially ordered set]] can be obtained from the MacNeille completion (cf. [[Completion, MacNeille (of a partially ordered set)|Completion, MacNeille (of a partially ordered set)]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005016.png" /> by removing its largest and smallest elements. The Dedekind completion is unique in the following sense. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005018.png" /> are Dedekind completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005019.png" />, then there exists a unique Riesz isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005021.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005023.png" />. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005024.png" /> is a Dedekind completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005025.png" />, then every normal Riesz homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005026.png" /> into any Dedekind-complete Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005027.png" /> can uniquely be extended to a normal Riesz homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005028.png" />.

The Riesz spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005030.png" />) are Dedekind complete; so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005032.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005033.png" />-finite. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005035.png" /> a compact [[Hausdorff space|Hausdorff space]]) is Dedekind complete if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005036.png" /> is extremally disconnected (cf. [[Extremally-disconnected space|Extremally-disconnected space]]). There are few non-trivial instances of Riesz spaces whose Dedekind completions are to some extent  "understood" . The Dedekind completion of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005037.png" /> of all converging sequences is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005038.png" />. That of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005039.png" /> is the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005041.png" /> is the space of all bounded Borel functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005042.png" /> is the ideal of all functions that vanish off meager sets (cf. [[Category of a set|Category of a set]]). (In either case, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110050/d11005043.png" /> is obvious.)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. de Jonge,   A. van Rooij,   "Introduction to Riesz spaces" , ''Tracts'' , '''8''' , Math. Centre, Amsterdam  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Luxemburg,   A.C. Zaanen,   "Riesz spaces" , '''I''' , North-Holland  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.Z. Vulikh,   "Introduction to the theory of partially ordered spaces" , Wolters–Noordhoff  (1967)  (In Russian)</TD></TR></table>