algebraic lattice

A latticeMathworldPlanetmath L is said to be an algebraic lattice if it is a complete latticeMathworldPlanetmath 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.


  1. 1.

    Groups. The lattice L(G) of subgroups of a group G is known to be completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Cyclic subgroups are compact elements of L(G). Since every subgroup H of G is the join of cyclic subgroups, each generated by an element hH, L(G) is algebraic.

  2. 2.

    Vector spaces. The lattice L(V) of subspacesMathworldPlanetmath of a vector space V is complete. Since each subspace has a basis, and since each element generates a one-dimensional subspace which is clearly compactPlanetmathPlanetmath, L(V) is algebraic.

  3. 3.

    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 rI. So L(R) is algebraic.

  4. 4.

    Modules. The above two examples can be combined and generalized into one, the lattice L(M) of submodules of a module M over a ring. The argumentsPlanetmathPlanetmath are similar.

  5. 5.

    Topological spacesMathworldPlanetmath. The lattice of closed subsets of a topological space is in general not algebraic. The simplest example is with the open intervals forming the subbasis. To begin with, it is not complete: the union of closed subsets [0,1-1n], n is [0,1), not a closed set. In additionPlanetmathPlanetmath, itself is a closed subset that is not compact.


  • 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=S, where SL is a set of compact elements sL, 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 S=0a), so aS and therefore a is compact.

  • 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 aD can be written as a=C, where C is a directed setMathworldPlanetmath (in D) such that each element in C is compact.


  • 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, ContinuousPlanetmathPlanetmath Lattices and Domains, Cambridge University Press, Cambridge (2003).
  • 4 S. Vickers, TopologyMathworldPlanetmath via Logic, Cambridge University Press, Cambridge (1989).
Title algebraic lattice
Canonical name AlgebraicLattice
Date of creation 2013-03-22 15:56:31
Last modified on 2013-03-22 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 compactly-generated lattice
Related topic SumOfIdeals
Defines algebraic dcpo