Distributivity

In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:

2 × (1 + 3) = (2 × 1) + (2 × 3).

In the left-hand side of the above equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the results added afterwards. Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3. Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Contents

Definition

Given a set S and two binary operations · and + on S, we say that the operation ·

x · (y + z) = (x · y) + (x · z);
(y + z) · x = (y · x) + (z · x);

Notice that when · is commutative, then the three above conditions are logically equivalent.

Examples

  1. Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers.
  2. Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.
  3. Matrix multiplication is distributive over matrix addition, even though not commutative.
  4. The union of sets is distributive over intersection, and intersection is distributive over union. Also, intersection is distributive over the symmetric difference.
  5. Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction. Also, conjunction is distributive over exclusive disjunction ("xor").
  6. For real numbers (or for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: max(a,min(b,c)) = min(max(a,b),max(a,c)) and min(a,max(b,c)) = max(min(a,b),min(a,c)).
  7. For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd(a,lcm(b,c)) = lcm(gcd(a,b),gcd(a,c)) and lcm(a,gcd(b,c)) = gcd(lcm(a,b),lcm(a,c)).
  8. For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) = min(a+b,a+c).

Distributivity and rounding

In practice, the distributive property of multiplication (and division) over addition is lost around the limits of arithmetic precision. For example, the identity ⅓+⅓+⅓ = (1+1+1)/3 appears to fail if conducted in decimal arithmetic; however many significant digits are used, the calculation will take the form 0.33333+0.33333+0.33333 = 0.99999 ≠ 1. Even where fractional numbers are representable exactly, errors will be introduced if rounding too far; for example, buying two books each priced at £14.99 before a tax of 17.5% in two separate transactions will actually save £0.01 over buying them together: £14.99×1.175 = £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98×1.175 = £35.23. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.

Distributivity in rings

Distributivity is most commonly found in rings and distributive lattices.

A ring has two binary operations (commonly called "+" and "*"), and one of the requirements of a ring is that * must distribute over +. Most kinds of numbers (example 1) and matrices (example 3) form rings. A lattice is another kind of algebraic structure with two binary operations, ∧ and ∨. If either of these operations (say ∧) distributes over the other (∨), then ∨ must also distribute over ∧, and the lattice is called distributive. See also the article on distributivity (order theory).

Examples 4 and 5 are Boolean algebras, which can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Examples 6 and 7 are distributive lattices which are not Boolean algebras.

Rings and distributive lattices are both special kinds of rigs, certain generalisations of rings. Those numbers in example 1 that don't form rings at least form rigs. Near-rigs are a further generalisation of rigs that are left-distributive but not right-distributive; example 2 is a near-rig.

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 implication operator of Heyting algebras. Details of 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 intervals.

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.

Notes

  1. ↑ Ayres, p. 20.

References

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