Difference between revisions of "Complete Dedekind lattice"
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| + | $#C+1 = 5 : ~/encyclopedia/old_files/data/C023/C.0203740 Complete Dedekind lattice | ||
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A complete lattice such that the identity | 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 | + | 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 [[#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> | ||
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====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) |
Complete Dedekind lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_Dedekind_lattice&oldid=46416