Distributivity (order theory)

In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.

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 (${\displaystyle \vee }$) and meet (${\displaystyle \wedge }$). 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

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