Distributivity

https://en.wikipedia.org/wiki/Distributive_property

Algebraic structures consisting of 2 binary operations (e.g. ring) are subject to distributivity of one operation over the other.

Distributivity ⨂ of over ⨁:

• left distributivity: a ⨂ (b ⨁ c) = (a ⨂ b) ⨁ (a ⨂ c)

• right distributivity: (b ⨁ c) ⨂ a = (b ⨂ a) ⨁ (c ⨂ a)

• (total) distributivity: left and right distributivity

Given a set $S$ and 2 binary operators $\star$ and $\odot$ on $S$, and $\forall x,y,z \in S$

• the operation $\star$ is left-distributive over $\odot$ if: $x*(y+z) = (x*y) + (x*z)$

• the operation $\star$ is right-distributive over $\odot$ if: $(y+z)*x=(y*x)+(z*x)$

• the operation $\star$ is distributive over $\odot$ if: it is both left and right distributive

• if $\star$ is commutative, these 3 axioms are equivalent: left-distributive ≡ distributive ≡ right-distributive

Algebraic structures consisting of 2 binary operations (e.g. ring) are subject to distributivity of one operation over the other.

Given a set S and 2 binary operators * and + on S:

• the operation * is left-distributive over + if

  $(\forall x,y,z \in S)\ .\ x*(y+z) = (x*y) + (x*z)$

• the operation * is right-distributive over + if

  $(\forall x,y,z \in S)\ .\ (y+z)*x=(y*x)+(z*x)$

• the operation * is distributive over + if it both left and right-distributive.

• when * is commutative, these 3 conditions are logically equivalent

a(b + c) = ab + ac

• The Distributive Law does not work for division