Definition:Distributive Operation
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Definition
Let $S$ be a set on which is defined two binary operations, defined on all the elements of $S \times S$, which we will denote as $\circ$ and $*$.
The operation $\circ$ is distributive over $*$, or distributes over $*$, if and only if:
- $\circ$ is right distributive over $*$
and:
- $\circ$ is left distributive over $*$.
Left Distributive
The operation $\circ$ is left distributive over the operation $*$ if and only if:
- $\forall a, b, c \in S: a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$
Right Distributive
The operation $\circ$ is right distributive over the operation $*$ if and only if:
- $\forall a, b, c \in S: \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$
So as to streamline what may turn into cumbersome language, some further definitions:
Distributand
Let $\circ$ be distributive over $*$.
Then $*$ is a distributand of $\circ$.
Distributor
Let $\circ$ be distributive over $*$.
Then $\circ$ is a distributor of $*$.
Also see
- Results about distributive operations can be found here.
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(c)}$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Operations
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): distributive law
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distributive
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distributive
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): distributive