View source for Complete lattice

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.

====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>



====Comments====
For the topic  "closure operation" , cf. also [[Closure relation|Closure relation]]; [[Basis|Basis]].