DistributivityIn 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)
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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.
Distributivity is most commonly found in ringss 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 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.