Complete lattice
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 |
Complete lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_lattice&oldid=33806