distributive law

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).

One of the laws relating to number operations. In symbols, it is stated: math.a(math.b + math.c) = math.amath.b + math.amath.c. The monomial factor math.a is distributed, or separately applied, to each term of the polynomial factor math.b + math.c, resulting in the product math.amath.b + math.amath.c. It can also be stated in words: The result of first adding several numbers and then multiplying the sum by some number is the same as first multiplying each separately by the number and then adding the products. Seealso associative law; commutative law.

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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 $(S,mu^S,eta^S)$ 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

$l:TSto\; ST$

such that the diagrams

         : and commute.

This law induces a composite monad ST with

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

See also

• distributive law

References

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