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distributive law. In mathematics, given any two operations, symbolized by * and ∘, the first operation, *, is distributive over the second, ∘, if *a**(*b*∘*c*)=(*a***b*)∘(*a***c*) for all possible choices of *a, b,* and *c.* Multiplication, ×, is distributive over addition, +, since for any numbers *a, b,* and *c, a*×(*b*+*c*)=(*a*×*b*)+(*a*×*c*). For example, for the numbers 2, 3, and 4, 2×(3+4)=14 and (2×3)+(2×4)=14, meaning that 2×(3+4)=(2×3)+(2×4). Strictly speaking, this law expresses only left distributivity, i.e., *a* is distributed from the left side of (*b*+*c*); the corresponding definition for right distributivity is (*a*+*b*)×*c*=(*a*×*c*)+(*b*×*c*).

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one.## See also

## References

Suppose that $(S,mu^S,eta^S)$ and $(T,mu^T,eta^T)$ are two monads on a category C. In general, there is no natural monad structure on the composite functor ST. On the other hand, there is a natural monad structure on the functor ST if there is a distributive law of the monad S over the monad T.

Formally, a distributive law of the monad S over the monad T is a natural transformation

- $l:TSto\; ST$

- : and commute.

This law induces a composite monad ST with

- as multiplication: $Smu^Tcdotmu^STTcdot\; SlT$,
- as unit: $eta^STcdoteta^T$.

- Jon Beck (1969). "Distributive laws".
*Lecture Notes in Mathematics*80 119–140. - Michael Barr and Charles Wells
*Toposes, Triples and Theories*. Springer-Verlag.

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