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Difference between revisions of "Complete lattice"

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====References====
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
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<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" , Hindushtan Publ. Comp.  (1977)  (Translated from Russian)</TD></TR>
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<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>
 
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<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</TD></TR>
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<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>
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<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>
 
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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=42455
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article