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interchange law
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(Definition)
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Let $S$ be a set and $\circ$ and $\bullet$ two partial binary operations on $S$ Then $\circ$ and $\bullet$ are said to satisfy the interchange law if $$(a\circ b)\bullet (c\circ d)=(a\bullet c)\circ (b\bullet d),$$ provided that the operations are defined on both sides of the equation.
An element $e\in S$ is a $\circ$ identity, or an identity with respect to $\circ$ if $e\circ a=a\circ e=a$ provided the operations are defined.
Proposition 1 If $\circ$ is a total function (defined for all of $S\times S$ , then there is at most one $\circ$ identity.
Proof. If $e$ and $f$ are both $\circ$ idenities, then $e=e\circ f=f$ 
Proposition 2 If both $\circ$ and $\bullet$ are total functions, and identities exist and the same with respect to both operations, then $\circ=\bullet$ and is commutative.
Proof. Suppose that $e$ is both the $\circ$ identity and the $\bullet$ identity. Then, according to the interchange law, $a\bullet d = (a\circ e)\bullet (e\circ d)=(a\bullet e)\circ (e\bullet d) = a \circ d$ showing that $\bullet = \circ$ Again, using the interchange law, $a\bullet d=(e\circ a)\bullet (d\circ e)= (e\bullet d)\circ (a\bullet e) = d\circ a = d\bullet a$ showing that $\bullet$ is commutative. 
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"interchange law" is owned by CWoo.
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Cross-references: commutative, total function, identity, equation, sides, operations, binary operations
There are 2 references to this entry.
This is version 2 of interchange law, born on 2008-09-30, modified 2008-10-01.
Object id is 11114, canonical name is InterchangeLaw.
Accessed 492 times total.
Classification:
| AMS MSC: | 08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous) | | | 18A99 (Category theory; homological algebra :: General theory of categories and functors :: Miscellaneous) |
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Pending Errata and Addenda
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