In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra.
- 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
of real numbers.
Given a set S and two binary operations * and +, it is said that
x * (y + z) = (x * y) + (x * z);
- * is left-distributive over + if, given any elements x, y, and z of S,
(y + z) * x = (y * x) + (z * x);
- * is right-distributive over + if, given any elements x, y, and z of S:
Notice that when * is commutative, then the three above conditions are logically equivalent.
- * is distributive over + if it is both left- and right-distributive.
Distributivity is most commonly found in ringss and distributive lattices.
- 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.
- Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.
- Matrix multiplication is distributive over matrix addition, even though it's not commutative.
- The union of sets is distributive over intersection, and intersection is distributive over union. Also, intersection is distributive over the symmetric difference.
- Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction. Also, conjunction is distributive over exclusive disjunction ("xor").
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 rigss, 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.
For sub-distributivity see Interval (mathematics)#Interval arithmetic.