distributiv

Distributiv, or distributive, is a term used in mathematics to describe a property of certain binary operations. Broadly, it refers to how one operation interacts with another when applied to a sum or join. In English, the standard term is distributive (property), while distributiv appears in some languages as the adjectival form.

In algebra, a binary operation distributes over another when it can be applied to a sum inside or outside the operation and yield the same result as distributing after the operation. Formally, a binary operation * distributes over + on the left if a*(b+c) = a*b + a*c for all a, b, c in a set. It distributes on the right if (a+b)*c = a*c + b*c for all a, b, c. If both hold, the operations are called distributive over each other. A common instance is multiplication over addition in the real numbers: a*(b+c) = ab + ac and (a+b)*c = ac + bc.

In logic and lattice theory, distributivity describes how two operations relate. For propositional logic, conjunction distributes over disjunction and, in many systems, disjunction distributes over conjunction. In lattice theory, a lattice is distributive if its join and meet operations satisfy the distributive laws: a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) and its dual.

Distributivity enables algebraic manipulation, factorization, and structured reasoning across mathematics, computer science, and logic.