Difference between revisions of "Complete lattice"
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| − | For the topic "closure | + | For the topic "closure operation" , cf. also [[Closure relation|Closure relation]]; [[Basis|Basis]]. |
Revision as of 15:24, 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.
Complete lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_lattice&oldid=17074