Definition
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 is distributive if the following additional identity holds for all x, y, and z in :
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:
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).
Read more about this topic: Distributive Lattice
Famous quotes containing the word definition:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)