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# distributive law

distributive law. In mathematics, given any two operations, symbolized by * and ∘, the first operation, *, is distributive over the second, ∘, if a*(bc)=(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+bc=(a×c)+(b×c).
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.

Suppose that $\left(S,mu^S,eta^S\right)$ and $\left(T,mu^T,eta^T\right)$ 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

$l:TSto ST$
such that the diagrams
: and commute.

This law induces a composite monad ST with

• as multiplication: $Smu^Tcdotmu^STTcdot SlT$,
• as unit: $eta^STcdoteta^T$.