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Difference between revisions of "Complete lattice"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR>
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====Comments====
 
====Comments====
 
For the topic  "closure operation" , cf. also [[Closure relation|Closure relation]]; [[Basis|Basis]].
 
For the topic  "closure operation" , cf. also [[Closure relation|Closure relation]]; [[Basis|Basis]].
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====References====
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<table>
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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>
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</table>

Revision as of 15:26, 18 October 2014

A partially ordered set in which any subset has a least upper bound and a greatest lower bound. These are usually called the join and the meet of and are denoted by and or simply by and (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 is complete if and only if any isotone mapping of the lattice into itself has a fixed point, i.e. an element such that . If is the set of subsets of a set ordered by inclusion and is a closure operation on , then the set of all -closed subsets is a complete lattice. Any partially ordered set can be isomorphically imbedded in a complete lattice, which in that case is called a completion of . 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=33804
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article