distributif

Distributif, in context, refers to the distributive property, a concept used in mathematics and related fields to describe how one operation interacts with another. The classic formulation says that a binary operation distributes over addition if, for all elements a, b, and c in a set, a⋅(b+c) = a⋅b + a⋅c, and (b+c)⋅a = b⋅a + c⋅a. When these equalities hold for all choices, the operation is said to be distributive over addition. The law also applies to subtraction, since a⋅(b−c) = a⋅b − a⋅c. Directions of distribution can be described as left-distributive or right-distributive if only one side holds.

In algebra, many standard structures feature distributivity as a defining property. Rings, fields, and semirings have two operations, typically addition and multiplication, with multiplication distributing over addition. This property underpins polynomial expansion, factorization, and many manipulations within these systems. Real numbers, integers, and polynomials form examples where the distributive law holds.

Beyond basic algebra, distributivity appears in other mathematical contexts. In lattice theory, a lattice is distributive if meet distributes over join and join distributes over meet. In classical propositional logic, conjunction distributes over disjunction and disjunction distributes over conjunction, aligning with the general idea of distributivity in logical operations.

The term is also used in linguistics to describe distributive readings, where actions are ascribed to each individual rather than to a whole group, with elements like “each,” “every,” or “per person” shaping the interpretation.

Etymology traces to Latin distributivus. In French, the term is distributif, while in English the related term is distributive.