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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:
4 · (2 + 3) = (4 · 2) + (4 · 3)
In the left-hand side of the above equation, the 4 multiplies the sum of 2 and 3; on the right-hand side, it multiplies the 2 and the 3 individually, with the results added afterwards. Because these give the same final answer (20), we say that multiplication by 4 distributes over addition of 2 and 3. Since we could have put any real numbers in place of 4, 2, and 3 above, and still gotten a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Definition

Given a set S and two binary operations * and +, it is said that

  • * is left-distributive over + if, given any elements x, y, and z of S,
x * (y + z) = (x * y) + (x * z);
  • * is right-distributive over + if, given any elements x, y, and z of S:
(y + z) * x = (y * x) + (z * x);
  • * is distributive over + if it is both left- and right-distributive.

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 it's 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").

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 v. If either of these operations (say ^) distributes over the other (v), then v must also distribute over ^, and the lattice is called distributive.

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.

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.



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