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Distributivity

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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.

How to Cite This Entry:
Distributivity. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Distributivity&oldid=37636
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article