Distributivity

In algebra, distributivity is a property of two binary operations which generalises the relationship between addition and multiplication in elementary algebra known as "multiplying out". For these elementary operations it is also known as the distributive law, expressed as

$a\times (b+c)=(a\times b)+(a\times c)$

Formally, let $\otimes$ and $\oplus$ be binary operations on a set X. We say that $\otimes$ left distributes over $\oplus$, or is left distributive, if

$a\otimes (b\oplus c)=(a\otimes b)\oplus (a\otimes c)\,$

and $\otimes$ right distributes over $\oplus$, or is right distributive, if

$(b\oplus c)\otimes a=(b\otimes a)\oplus (c\otimes a).\,$

The laws are of course equivalent if the operation $\otimes$ is commutative.

Examples

• In a ring, the multiplication distributes over the addition.
• In a vector space, multiplication by scalars distributes over addition of vectors.
• There are three closely connected examples where each of two operations distributes over the other:
  • In set theory, intersection distributes over union and union distributes over intersection;
  • In propositional logic, conjunction (logical and) distributes over disjunction (logical or) and disjunction distributes over conjunction;
  • In a distributive lattice, join distributes over meet and meet distributes over join.