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''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  $  L $, 
a Dedekind completion of  $  L $
is a pair  $  ( M,T ) $
where  $  M $
is a Riesz space and  $  T : L \rightarrow M $
is a mapping such that

1)  $  M $
is Dedekind complete;

2)  $  T $
is a Riesz isomorphism of  $  L $
onto a Riesz subspace  $  T ( L ) $
of  $  M $;

3) as a mapping  $  L \rightarrow M $, 
$  T $
is normal, i.e., it preserves arbitrary suprema and infima;

4) for all  $  a \in M $,

$$ 
a =  \sup  \left \{ {x \in T ( L ) } : {x \leq  a } \right \} =  \inf  \left \{ {x \in T ( L ) } : {x \geq  a } \right \} .
$$

Every Archimedean Riesz space  $  L $
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  $  L $
by removing its largest and smallest elements. The Dedekind completion is unique in the following sense. If  $  ( M _ {1} ,T _ {1} ) $
and  $  ( M _ {2} ,T _ {2} ) $
are Dedekind completions of  $  L $, 
then there exists a unique Riesz isomorphism  $  S $
of  $  M _ {1} $
onto  $  M _ {2} $
with  $  T _ {2} = S \circ T _ {1} $. 
More generally, if  $  ( M,T ) $
is a Dedekind completion of  $  L $, 
then every normal Riesz homomorphism of  $  L $
into any Dedekind-complete Riesz space  $  K $
can uniquely be extended to a normal Riesz homomorphism  $  M \rightarrow K $.

The Riesz spaces  $  L _ {p} ( \mu ) $(
$  1 \leq  p < \infty $) 
are Dedekind complete; so is  $  L _  \infty  ( \mu ) $
if  $  \mu $
is  $  \sigma $-
finite. The space  $  C ( X ) $(
$  X $
a compact [[Hausdorff space|Hausdorff space]]) is Dedekind complete if and only if  $  X $
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  $  c $
of all converging sequences is  $  l _  \infty  $. 
That of  $  C ( X ) $
is the quotient  $  B ( X ) /N $, 
where  $  B ( X ) $
is the space of all bounded Borel functions and  $  N $
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  $  T : L \rightarrow M $
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>