Generalizations of Distributivity
In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only one binary operation, such as the according definitions and their relations are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice.
In the presence of an ordering relation, one can also weaken the above equalities by replacing = by either ≤ or ≥. Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic.
In category theory, if (S, μ, η) and (S', μ', η') are monads on a category C, a distributive law S.S' → S'.S is a natural transformation λ : S.S' → S'.S such that (S', λ) is a lax map of monads S → S and (S, λ) is a colax map of monads S' → S' . This is exactly the data needed to define a monad structure on S'.S: the multiplication map is S'μ.μ'S².S'λS and the unit map is η'S.η. See: distributive law between monads.
A generalized distributive law has also been proposed in the area of information theory.
Read more about this topic: Distributive Property