Distributivity

distributivity law, distributive property, of one operation with respect to another

The property of a pair of binary algebraic operations (cf. Algebraic operation), expressed by one of the following identities: $$ (D1)\ \ \ \ \forall x,y,z\ \ x \otimes (y \oplus z) = (x \otimes y) \oplus (x \otimes z) $$ $$ (D2)\ \ \ \ \forall x,y,z\ \ (x \oplus y) \otimes z = (x \otimes z) \oplus (y \otimes z) $$ where $\oplus, \otimes$ are the symbols of the binary operations, and $x,y,z$ are object variables. If in a set $A$ two specific binary operations $+, \circ$ are defined, i.e. two mappings $$ + : A \times A \rightarrow A \ ,\ \ \ \circ : A \times A \rightarrow A $$

are given, and the symbols $\oplus, \otimes$ are interpreted as the symbols of the respective operations $+, \circ$ in $A$, one can speak of the truth or falsehood of each one of the formulas D1 and D2 in $A$. If both these formulas are true in $A$, the operation $\circ$ is called distributive with respect to the operation $+$ in $A$.

Comments

I.e., in $A$ the operation $\circ$ is distributive with respect to $+$ if for all $a,b,c \in A$ one has $a \circ (b+c) = (a \circ b) + (a \circ c)$ and $(a + b) \circ c = (a \circ c) + (b \circ c)$. It may also be expressed as $\circ$ "distributes over" $+$.

For example, multiplication is distributive with respect to addition in the set of real numbers and in the set of integers. A distributive lattice is a lattice in which one of the operations $\vee$ (join) and $\wedge$ (meet) is distributive over the other; in this case if one of the laws holds, so does the other. For example, each of the operations union of sets and intersection of sets distributes over the other.