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Complete lattice

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2020 Mathematics Subject Classification: Primary: 06B23 [MSN][ZBL]

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 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. For example, the map x \mapsto \{x\}^\nabla = \{ y \in X : y \le x \} maps P into the complete lattice \mathcal{P}(P), and hence defines a completion \mathcal{O}(P) of P. However this has this disadvantage that if P is already a complete lattice, and hence a completion of itself, then the completion \mathcal{O}(P) is larger than P itself. The Dedkind–MacNeille completion 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 (note that while the meet of a family of closed sets is their set-theoretic intersection, the join of a family of closed sets is the closure of their set-theoretic union).

References

[1] G. Birkhoff, "Lattice theory", 3rd ed. Colloq. Publ. , 25 , Amer. Math. Soc. (1967) Zbl 0153.02501
[2] L.A. Skornyakov, "Elements of lattice theory" , Hindustan Publ. Comp. (1977) (Translated from Russian) ISBN 0852743319 Zbl 0222.06002
[a1] B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1 Zbl 1002.06001
[a2] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.V. Mislove, D.S. Scott, "A compendium of continuous lattices" , Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
How to Cite This Entry:
Complete lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_lattice&oldid=54596
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article