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Difference between revisions of "Cancellation law"

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(Start article: Cancellation law)
 
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In an algebraic structure $A$ with a [[binary operation]] $\cdot$, the left and right cancellation laws respectively hold if for all $x,y,x$
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In an algebraic structure $A$ with a [[binary operation]] $\cdot$, the left and right cancellation laws respectively hold if for all $x,y,z$
 
$$
 
$$
 
x \cdot y = x \cdot z \Rightarrow y = z \,,
 
x \cdot y = x \cdot z \Rightarrow y = z \,,
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x \cdot y = z \cdot y \Rightarrow x = z \ .
 
x \cdot y = z \cdot y \Rightarrow x = z \ .
 
$$
 
$$
Such a structure is termed "cancellative"
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Such a structure is termed "cancellative".
  
 
A [[group]] satisfies the cancellation laws; a finite [[semi-group]] is a group if and only if it satisfies the cancellation laws.  A commutative semi-group embeds in a group if and only if it is cancellative: for non-commutative groups the analogous statement does not hold in general, see [[Imbedding of semi-groups]].
 
A [[group]] satisfies the cancellation laws; a finite [[semi-group]] is a group if and only if it satisfies the cancellation laws.  A commutative semi-group embeds in a group if and only if it is cancellative: for non-commutative groups the analogous statement does not hold in general, see [[Imbedding of semi-groups]].
  
A [ring]] is an [[integral domain]] if it is commutative and satisfies the cancellation laws for non-zero elements,
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A [[ring]] is an [[integral domain]] if it is commutative and satisfies the cancellation laws for non-zero elements,

Latest revision as of 18:23, 3 January 2016

In an algebraic structure $A$ with a binary operation $\cdot$, the left and right cancellation laws respectively hold if for all $x,y,z$ $$ x \cdot y = x \cdot z \Rightarrow y = z \,, $$ $$ x \cdot y = z \cdot y \Rightarrow x = z \ . $$ Such a structure is termed "cancellative".

A group satisfies the cancellation laws; a finite semi-group is a group if and only if it satisfies the cancellation laws. A commutative semi-group embeds in a group if and only if it is cancellative: for non-commutative groups the analogous statement does not hold in general, see Imbedding of semi-groups.

A ring is an integral domain if it is commutative and satisfies the cancellation laws for non-zero elements,

How to Cite This Entry:
Cancellation law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cancellation_law&oldid=35787