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A [[Partially ordered set|partially ordered set]] in which any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237901.png" /> has a least upper bound and a greatest lower bound. These are usually called the join and the meet of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237902.png" /> and are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237904.png" /> or simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237906.png" /> (respectively). If a partially ordered set has a largest element and each non-empty subset of it has a greatest lower bound, then it is a complete lattice. A lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237907.png" /> is complete if and only if any isotone mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237908.png" /> of the lattice into itself has a fixed point, i.e. an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c0237909.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379011.png" /> is the set of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379012.png" /> ordered by inclusion and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379013.png" /> is a closure operation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379014.png" />, then the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379015.png" />-closed subsets is a complete lattice. Any partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379016.png" /> can be isomorphically imbedded in a complete lattice, which in that case is called a completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023790/c02379017.png" />. The completion by sections (cf. [[Completion, MacNeille (of a partially ordered set)|Completion, MacNeille (of a partially ordered set)]]) is the least of all completions of a given partially ordered set. Complete lattices are formed by the set of all subalgebras in a universal algebra, by the set of all congruences in a universal algebra, and by the set of all closed subsets in a topological space.
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A [[Partially ordered set|partially ordered set]] in which any subset $A$ has a least upper bound and a greatest lower bound. These are usually called the join and the meet of $A$ and are denoted by $\Wedge_{a \in A} a$ and  and $\Vee_{a \in A} a$ or simply by $\vee A$ and $\wedge A$ (respectively). If a partially ordered set has a largest element and each non-empty subset of it has a greatest lower bound, then it is a complete lattice. A lattice $L$ is complete if and only if any isotone mapping $\phi$ of the lattice into itself has a fixed point, i.e. an element $a \in L$ such that $a \phi = a$. If $\mathcal{P}(M)$ is the set of subsets of a set $M$ ordered by inclusion and $\phi$ is a closure operation on $\mathcal{P}(M)$, then the set of all $\phi$-closed subsets is a complete lattice. Any partially ordered set $P$ can be isomorphically imbedded in a complete lattice, which in that case is called a completion of $P$. The completion by sections (cf. [[Completion, MacNeille (of a partially ordered set)|Completion, MacNeille (of a partially ordered set)]]) is the least of all completions of a given partially ordered set. Complete lattices are formed by the set of all subalgebras in a universal algebra, by the set of all congruences in a universal algebra, and by the set of all closed subsets in a topological space.
  
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) ISBN 978-0-521-78451-1</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) ISBN 978-0-521-78451-1</TD></TR>
 
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Revision as of 15:31, 18 October 2014

A partially ordered set in which any subset $A$ has a least upper bound and a greatest lower bound. These are usually called the join and the meet of $A$ and are denoted by $\Wedge_{a \in A} a$ and and $\Vee_{a \in A} a$ or simply by $\vee A$ and $\wedge A$ (respectively). If a partially ordered set has a largest element and each non-empty subset of it has a greatest lower bound, then it is a complete lattice. A lattice $L$ is complete if and only if any isotone mapping $\phi$ of the lattice into itself has a fixed point, i.e. an element $a \in L$ such that $a \phi = a$. If $\mathcal{P}(M)$ is the set of subsets of a set $M$ ordered by inclusion and $\phi$ is a closure operation on $\mathcal{P}(M)$, then the set of all $\phi$-closed subsets is a complete lattice. Any partially ordered set $P$ can be isomorphically imbedded in a complete lattice, which in that case is called a completion of $P$. The completion by sections (cf. Completion, MacNeille (of a partially ordered set)) is the least of all completions of a given partially ordered set. Complete lattices are formed by the set of all subalgebras in a universal algebra, by the set of all congruences in a universal algebra, and by the set of all closed subsets in a topological space.

References

[1] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2] L.A. Skornyakov, "Elements of lattice theory" , Hindushtan Publ. Comp. (1977) (Translated from Russian)


Comments

For the topic "closure operation" , cf. also Closure relation; Basis.

References

[a1] B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1
How to Cite This Entry:
Complete lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_lattice&oldid=33806
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article