distributives

Distributivity is a structural property that relates two binary operations in a mathematical system, allowing one operation to distribute over the other. The classic form is the distributive law a*(b+c) = ab + ac, with the dual (a+b)*c = ac + bc, which expresses how multiplication interacts with addition.

In algebra, distributivity is central to the definition of rings and related structures. Multiplication distributes over addition in rings, fields, and modules, ensuring that expressions like a*(b+c) and (a+b)*c expand predictably. This property holds for familiar systems such as integers, real numbers, and matrices (where matrix multiplication distributes over matrix addition).

In lattice theory, distributivity concerns the meet (often written as ∧) and join (∨) operations. A lattice is distributive if x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and, dually, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Boolean algebras are canonical examples of distributive lattices, reinforcing a compatible interaction between the two basic operations.

In logic, distributivity describes how conjunction (and) and disjunction (or) interact. In classical propositional logic, P ∧ (Q ∨ R) is equivalent to (P ∧ Q) ∨ (P ∧ R), and P ∨ (Q ∧ R) is equivalent to (P ∨ Q) ∧ (P ∨ R). Some non-classical logics do not satisfy one or both of these distributive laws.

Not all mathematical structures are distributive; there exist lattices and algebras in which the distributive laws fail. Recognizing distributivity helps classify systems and informs the expected behavior of combined operations.