distributivity

Distributivity is a property of binary operations in mathematics and logic describing how one operation interacts with another when combining multiple elements. In its most common form, an operation distributes over another if applying the first operation to a sum or join yields the same result as summing or joining the results of the operation applied to each term separately.

Formally, if two binary operations are defined on a set S, the operation ⊗ distributes over ⊕ when a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c) for all a, b, c in S. If the equality also holds when the order is reversed, i.e., (b ⊕ c) ⊗ a = (b ⊗ a) ⊕ (c ⊗ a), then ⊗ distributes on both the left and the right. Some structures require only one-sided distributivity, while others have both.

Examples include standard arithmetic, where multiplication distributes over addition: a × (b + c) = ab + ac. In matrix algebra, matrix multiplication distributes over addition: A(B + C) = AB + AC and (B + C)A = BA + CA. In vector spaces, scalar multiplication distributes over vector addition: α(u + v) = αu + αv. In algebraic structures such as rings, multiplication distributes over addition, which is part of the ring axioms.

Not all algebraic systems are distributive. Some lattices, for instance, are non-distributive, meaning their join and meet operations do not satisfy the distributive laws. Distributivity also appears in broader contexts, including category theory and universal algebra, where it expresses how certain operations interact.

See also: Associativity, Commutativity, Ring theory, Lattice theory.