Namespaces
Variants
Actions

Difference between revisions of "Complete lattice"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(→‎References: expand bibliodata)
 
(7 intermediate revisions by the same user not shown)
Line 1: Line 1:
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.
+
{{TEX|done}}{{MSC|06B23}}
  
====References====
+
A [[partially ordered set]] in which any subset $A$ 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 relation|closure operation]] on $\mathcal{P}(M)$, then the set of all $\phi$-closed subsets is a complete lattice.  
<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>
+
 
 +
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 [[Completion, MacNeille (of a partially ordered set)|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====
 +
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory", 3rd ed. ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1967)  {{ZBL|0153.02501}}</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , Hindustan Publ. Comp.  (1977)  (Translated from Russian) ISBN 0852743319 {{ZBL|0222.06002}}</TD></TR>
 +
</table>
  
====Comments====
+
<table>
For the topic "closure operationclosure operation" , cf. also [[Closure relation|Closure relation]]; [[Basis|Basis]].
+
<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 {{ZBL|1002.06001}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> 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  {{MR|0614752}}  {{ZBL|0452.06001}} </TD></TR>
 +
</table>

Latest revision as of 17:39, 9 December 2017

2010 Mathematics Subject Classification: Primary: 06B23 [MSN][ZBL]

A partially ordered set in which any subset $A$ 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=17074
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article