Distributivní

Distributivní, in English distributive, is a mathematical term describing a property of how two operations interact. An operation D is said to be distributive over another operation O if applying D to a combination of operands can be rewritten as a combination of applications of D to the operands separately. The term is used across arithmetic, algebra, logic and set theory.

In arithmetic, the best-known example is the distributive property of multiplication over addition: a × (b + c) = a × b + a × c for all numbers a, b, c. A similar rule holds for (b + c) × a. This property also extends to subtraction: a × (b − c) = a × b − a × c. Not all operations distribute; division does not distribute over addition, and addition does not distribute over multiplication.

In logic and Boolean algebra, distributive laws describe how conjunction (and) and disjunction (or) interact: p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r), and p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r). When these laws hold for all statements, the logic or algebra is said to be distributive.

In lattice theory, a distributive lattice is one where meet distributes over join and join distributes over meet. A classic example is the lattice of subsets of a set with union and intersection: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). There are non-distributive lattices as well, such as certain lattices of subspaces of a vector space.

Originating from Latin distributivus, the term is used primarily in mathematics to characterize how one operation distributes over another, guiding both proofs and structural properties.