Distributive

Distributive describes a property of certain binary operations in mathematics, where one operation can be distributed over another according to a distribution law. The most familiar form is the distributive law of multiplication over addition: a(b + c) = ab + ac, and equivalently (a + b)c = ac + bc. This holds in standard arithmetic and is a foundational axiom in ring theory, where multiplication distributes over addition on both the left and the right.

In logic, distributivity describes how connectives interact: p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) and p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r). Some logics or lattices are non-distributive and do not satisfy these laws.

In order theory, a lattice is distributive if meet distributes over join and join distributes over meet. Boolean algebras are distributive lattices and model classical logic. The smallest nondistributive lattices are the diamond M3 and the pentagon N5, which illustrate the limits of distributivity.

Distributivity also appears in set theory, where union distributes over intersection and intersection distributes over union: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).