Difference between revisions of "Cancellation law"
From Encyclopedia of Mathematics
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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, | + | In an algebraic structure $A$ with a [[binary operation]] $\cdot$, the left and right cancellation laws respectively hold if for all $x,y,z$ |
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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 \ . | ||
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| − | Such a structure is termed "cancellative" | + | 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, | + | 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
Cancellation law. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cancellation_law&oldid=35787