PlanetMath: distributivity

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distributivity

(Definition)

Given a set with two binary operations $+\colon S \times S \to S$ and $\cdot\colon S \times S \to S$ , we say that $\cdot$ is right distributive over if

$\displaystyle (a+b) \cdot c = (a \cdot c) + (b \cdot c)\mathrm{~for~all~} a,b,c \in S$

and left distributive over

$\displaystyle a \cdot (b+c) = (a \cdot b) + (a \cdot c)\mathrm{~for~all~}a,b,c \in S.$

If $\cdot$ is both left and right distributive over

, then it is said to be distributive over

(or, alternatively, we may say that $\cdot$ distributes over

"distributivity" is owned by yark.

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See Also: ring, distributive lattice, near-ring

Other names:

distributive law, distributive property

Also defines:

distributive, left distributive, right distributive, left-distributive, right-distributive, distributes over, left distributivity, right distributivity, left distributes over, left distributive law, right distributive law

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Cross-references: binary operations
There are 72 references to this entry.

This is version 12 of distributivity, born on 2003-07-22, modified 2007-06-29.
Object id is 4493, canonical name is Distributive.
Accessed 15597 times total.

Classification:

AMS MSC:	06D99 (Order, lattices, ordered algebraic structures :: Distributive lattices :: Miscellaneous)
	13-00 (Commutative rings and algebras :: General reference works )
	16-00 (Associative rings and algebras :: General reference works )
	17-00 (Nonassociative rings and algebras :: General reference works )

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