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.

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