Distributive Lattices
Probably the most common type of distributivity is the one defined for lattices, where the formation of binary suprema and infima provide the total operations of join and meet . Distributivity of these two operations is then expressed by requiring that the identity
hold for all elements x, y, and z. This distributivity law defines the class of distributive lattices. Note that this requirement can be rephrased by saying that binary meets preserve binary joins. The above statement is known to be equivalent to its order dual
such that one of these properties suffices to define distributivity for lattices. Typical examples of distributive lattice are totally ordered sets, Boolean algebras, and Heyting algebras.
Read more about this topic: Distributivity (order Theory)