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
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