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Difference between revisions of "Complete Dedekind lattice"

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A complete lattice such that the identity
 
A complete lattice such that 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/c023740/c0237401.png" /></td> </tr></table>
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$$
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\left ( \wedge _ {i \in I } a _ {i} \right ) \
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\wedge  \left ( \lor _ {i \in I } b _ {i} \right )  = \
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\lor _ {i \in I }  ( a _ {i}  \wedge  b _ {i} )
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$$
  
is valid for any of its elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023740/c0237402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023740/c0237403.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023740/c0237404.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023740/c0237405.png" />. Any complete Dedekind lattice is a modular lattice. If a universal algebra has commuting congruences, then its congruence lattice is a complete Dedekind lattice [[#References|[1]]].
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is valid for any of its elements $  a _ {i} , b _ {i} $,  
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$  i \in I $,  
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for which $  a _ {i} \geq  b _ {j} $
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if $  i \neq j $.  
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Any complete Dedekind lattice is a modular lattice. If a universal algebra has commuting congruences, then its congruence lattice is a complete Dedekind lattice [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ph. Dwinger,  "Some theorems on universal algebras III"  ''Indag. Math.'' , '''20'''  (1958)  pp. 70–76</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Ph. Dwinger,  "Some theorems on universal algebras III"  ''Indag. Math.'' , '''20'''  (1958)  pp. 70–76</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 17:45, 4 June 2020


A complete lattice such that the identity

$$ \left ( \wedge _ {i \in I } a _ {i} \right ) \ \wedge \left ( \lor _ {i \in I } b _ {i} \right ) = \ \lor _ {i \in I } ( a _ {i} \wedge b _ {i} ) $$

is valid for any of its elements $ a _ {i} , b _ {i} $, $ i \in I $, for which $ a _ {i} \geq b _ {j} $ if $ i \neq j $. Any complete Dedekind lattice is a modular lattice. If a universal algebra has commuting congruences, then its congruence lattice is a complete Dedekind lattice [1].

References

[1] Ph. Dwinger, "Some theorems on universal algebras III" Indag. Math. , 20 (1958) pp. 70–76

Comments

The term "Dedekind lattice" is seldom used in the English language literature, instead one uses modular lattice (cf. [a1]). However, a complete modular lattice is a complete lattice satisfying the (finite) modular law. The notion defined in the article above has no established name; it could be called a completely modular lattice.

References

[a1] P.M. Cohn, "Universal algebra" , Reidel (1981)
How to Cite This Entry:
Complete Dedekind lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_Dedekind_lattice&oldid=46416
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article