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A [[Complete lattice|complete lattice]] in which the identity
 
A [[Complete lattice|complete lattice]] in which the identity
  
<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/c/c023/c023960/c0239601.png" /></td> </tr></table>
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$$
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\inf _ {i \in I } \
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\sup _ {j \in J _ {i} } \
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a _ {i,j}  = \
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\sup _ {f \in F } \
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\inf _ {i \in I } \
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a _ {i, f ( i) }
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$$
  
(called the complete distributive law) holds for all doubly-indexed families of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239602.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239603.png" /> is the set of all choice functions for the family of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239604.png" />. Like the finite distributive law (see [[Distributive lattice|Distributive lattice]]), the complete distributive law is equivalent to its dual; that is, a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239605.png" /> is completely distributive if and only if the opposite lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239606.png" /> is completely distributive. Completely distributive lattices may also be characterized as those complete lattices in which every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239607.png" /> is expressible as the supremum of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239608.png" /> such that, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c0239609.png" /> is any subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c02396010.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c02396011.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c02396012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023960/c02396013.png" /> [[#References|[a1]]]. Any complete [[Totally ordered set|totally ordered set]] is completely distributive; a complete [[Boolean algebra|Boolean algebra]] is completely distributive if and only if it is isomorphic to the full power set of some set. In general, a complete lattice is completely distributive if and only if it is imbeddable in a full power set by a mapping preserving arbitrary sups and infs.
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(called the complete distributive law) holds for all doubly-indexed families of elements $  \{ {a _ {i,j} } : {i \in I,  j \in J } \} $,  
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where $  F $
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is the set of all choice functions for the family of sets $  \{ {J _ {i} } : {i \in I } \} $.  
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Like the finite distributive law (see [[Distributive lattice|Distributive lattice]]), the complete distributive law is equivalent to its dual; that is, a lattice $  A $
 +
is completely distributive if and only if the opposite lattice $  A ^ { \mathop{\rm op} } $
 +
is completely distributive. Completely distributive lattices may also be characterized as those complete lattices in which every element $  a $
 +
is expressible as the supremum of elements $  b $
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such that, whenever $  S $
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is any subset of $  A $
 +
with $  \sup  S \geq  a $,  
 +
there exists an $  s \in S $
 +
with $  s \geq  b $[[#References|[a1]]]. Any complete [[Totally ordered set|totally ordered set]] is completely distributive; a complete [[Boolean algebra|Boolean algebra]] is completely distributive if and only if it is isomorphic to the full power set of some set. In general, a complete lattice is completely distributive if and only if it is imbeddable in a full power set by a mapping preserving arbitrary sups and infs.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.N. Raney,  "Completely distributive complete lattices"  ''Proc. Amer. Math. Soc.'' , '''3'''  (1952)  pp. 677–680</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.N. Raney,  "Completely distributive complete lattices"  ''Proc. Amer. Math. Soc.'' , '''3'''  (1952)  pp. 677–680</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1967)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


A complete lattice in which the identity

$$ \inf _ {i \in I } \ \sup _ {j \in J _ {i} } \ a _ {i,j} = \ \sup _ {f \in F } \ \inf _ {i \in I } \ a _ {i, f ( i) } $$

(called the complete distributive law) holds for all doubly-indexed families of elements $ \{ {a _ {i,j} } : {i \in I, j \in J } \} $, where $ F $ is the set of all choice functions for the family of sets $ \{ {J _ {i} } : {i \in I } \} $. Like the finite distributive law (see Distributive lattice), the complete distributive law is equivalent to its dual; that is, a lattice $ A $ is completely distributive if and only if the opposite lattice $ A ^ { \mathop{\rm op} } $ is completely distributive. Completely distributive lattices may also be characterized as those complete lattices in which every element $ a $ is expressible as the supremum of elements $ b $ such that, whenever $ S $ is any subset of $ A $ with $ \sup S \geq a $, there exists an $ s \in S $ with $ s \geq b $[a1]. Any complete totally ordered set is completely distributive; a complete Boolean algebra is completely distributive if and only if it is isomorphic to the full power set of some set. In general, a complete lattice is completely distributive if and only if it is imbeddable in a full power set by a mapping preserving arbitrary sups and infs.

References

[a1] G.N. Raney, "Completely distributive complete lattices" Proc. Amer. Math. Soc. , 3 (1952) pp. 677–680
[a2] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)
How to Cite This Entry:
Completely distributive lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely_distributive_lattice&oldid=46425