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Concept
Distributive lattice
Summary
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.
As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient.
A lattice (L,∨,∧) is distributive if the following additional identity holds for all x, y, and z in L:
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).
Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins. It is a basic fact of lattice theory that the above condition is equivalent to its dual:
x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) for all x, y, and z in L.
In every lattice, defining p≤q as usual to mean p∧q=p, the inequality x ∧ (y ∨ z) ≥ (x ∧ y) ∨ (x ∧ z) holds as well as its dual inequality x ∨ (y ∧ z) ≤ (x ∨ y) ∧ (x ∨ z). A lattice is distributive if one of the converse inequalities holds, too.
More information on the relationship of this condition to other distributivity conditions of order theory can be found in the article on distributivity (order theory).
A morphism of distributive lattices is just a lattice homomorphism as given in the article on lattices, i.e. a function that is compatible with the two lattice operations. Because such a morphism of lattices preserves the lattice structure, it will consequently also preserve the distributivity (and thus be a morphism of distributive lattices).
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.
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