Distributivitat

Distributivitat, or distributivity, is a property that describes how two binary operations interact. If an operation, typically multiplication, distributes over addition, it satisfies the distributive laws: a×(b+c) = a×b + a×c for all elements a, b, c; and, when applicable, (a+b)×c = a×c + b×c. If both laws hold, the operation distributes over addition on both sides. If only one side holds, the operation is called left- or right-distributive.

In arithmetic and algebra, multiplication distributes over addition on familiar sets such as the real numbers, integers, polynomials, and matrices. This means expressions can be rewritten by applying the distributive laws, facilitating simplification and factorization. For example, 3×(4+5) equals 3×4 + 3×5, giving 12 + 15 = 27.

In logic and Boolean algebra, similar distributive laws apply to connectives. Conjunction distributes over disjunction: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). In Boolean algebras, both distributive laws hold, making them distributive lattices.

Distributivity is a fundamental axiom in many algebraic structures, such as rings, where multiplication distributes over addition. It also helps define and analyze factorization, polynomial identities, and module theory.

Not all algebraic structures are distributive. Some lattices, such as the lattice of subspaces of a vector space with join and meet interpreted as sum and intersection, are not distributive in general. The property of distributivity thus distinguishes different types of algebraic systems and guides both theory and computation.