Distributively

Distributively is an adverb used to describe the manner in which operations interact under the distributive law. In mathematics and related fields, a binary operation is said to distribute over another when applying it to a sum, union, or join yields the same result as applying it to each addend, then combining the results. When an operation is distributive over another, it is common to describe it as distributing across that operation or operating distributively over it.

In elementary arithmetic, multiplication distributes over addition: a*(b+c) = a*b + a*c; and similarly (b+c)*a = b*a + c*a. In algebraic structures such as rings, multiplication distributes over addition: x*(y+z) = x*y + x*z (and (y+z)*x = y*x + z*x). In logic and Boolean algebra, conjunction distributes over disjunction and vice versa: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r), and p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r). In set theory, union and intersection distribute: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

In lattice theory, a distributive lattice is one in which the meet and join operations distribute over each other. Some algebraic structures are not distributive, and their lattices of elements fail one or both distributive laws, exemplified by certain nondistributive lattices such as M3 or N5.

Distributivity is a foundational principle that enables factorization, expansion, and simplification across many branches of mathematics, computer science, and logic. The term distributively often appears in formal statements and proofs describing this property.