Distributive Lattice - Examples

Examples

Distributive lattices are ubiquitous but also rather specific structures. As already mentioned the main example for distributive lattices are lattices of sets, where join and meet are given by the usual set-theoretic operations. Further examples include:

  • The Lindenbaum algebra of most logics that support conjunction and disjunction is a distributive lattice, i.e. "and" distributes over "or" and vice versa.
  • Every Boolean algebra is a distributive lattice.
  • Every Heyting algebra is a distributive lattice. Especially this includes all locales and hence all open set lattices of topological spaces. Also note that Heyting algebras can be viewed as Lindenbaum algebras of intuitionistic logic, which makes them a special case of the above example.
  • Every totally ordered set is a distributive lattice with max as join and min as meet.
  • The natural numbers form a distributive lattice (complete as a meet-semilattice) with the greatest common divisor as meet and the least common multiple as join.
  • Given a positive integer n, the set of all positive divisors of n forms a distributive lattice, again with the greatest common divisor as meet and the least common multiple as join. This is a Boolean algebra if and only if n is square-free.
  • A lattice-ordered vector space is a distributive lattice.
  • Young's lattice given by the inclusion ordering of Young diagrams representing integer partitions is a distributive lattice.

Read more about this topic:  Distributive Lattice

Famous quotes containing the word examples:

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)