algebraic lattice
A lattice^{} $L$ is said to be an algebraic lattice if it is a complete lattice^{} and every element of $L$ can be written as a join of compact elements.
As the name (G. Birkhoff originally coined the term) suggests, algebraic lattices are mostly found in lattices of subalgebras of algebraic systems. Below are some common examples.
Examples.
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Rings. The lattice $L(R)$ of ideals of a ring $R$ is also complete, the join of a set of ideals of $R$ is the ideal generated by elements in each of the ideals in the set. Any ideal $I$ is the join of cyclic ideals generated by elements $r\in I$. So $L(R)$ is algebraic.
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Topological spaces^{}. The lattice of closed subsets of a topological space is in general not algebraic. The simplest example is $\mathbb{R}$ with the open intervals forming the subbasis. To begin with, it is not complete: the union of closed subsets $[0,1\frac{1}{n}]$, $n\in \mathbb{N}$ is $[0,1)$, not a closed set. In addition^{}, $\mathbb{R}$ itself is a closed subset that is not compact.
Remarks.

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Since every element in an algebraic lattice is a join of compact elements, it is easy to see that every atom is compact: for if $a$ is an atom in an algebraic lattice $L$, and $a=\bigvee S$, where $S\subseteq L$ is a set of compact elements $s\in L$, then each $s$ is either $0$ or $a$. Therefore, $S$ consists of at most two elements $0$ and $a$. But $S$ can’t be a singleton consisting of $0$ (otherwise $\bigvee S=0\ne a$), so $a\in S$ and therefore $a$ is compact.

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The notion of being algebraic in a lattice can be generalized to an arbitrary dcpo: an algebraic dcpo is a dcpo $D$ such that every $a\in D$ can be written as $a=\bigvee C$, where $C$ is a directed set^{} (in $D$) such that each element in $C$ is compact.
References
 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
 3 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous^{} Lattices and Domains, Cambridge University Press, Cambridge (2003).
 4 S. Vickers, Topology^{} via Logic, Cambridge University Press, Cambridge (1989).
Title  algebraic lattice 

Canonical name  AlgebraicLattice 
Date of creation  20130322 15:56:31 
Last modified on  20130322 15:56:31 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  15 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B23 
Classification  msc 51D25 
Synonym  compactlygenerated lattice 
Related topic  SumOfIdeals 
Defines  algebraic dcpo 