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cancellative semigroup
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(Definition)
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Let $S$ be a semigroup.
$S$ is left cancellative if, for all $a,b,c\in S$ $ab=ac\Rightarrow b=c$ $S$ is right cancellative if, for all $a,b,c\in S$ $ba=ca\Rightarrow b=c$ $S$ is cancellative if it is both left and right cancellative.
This is a generalisation of groups, and in fact being cancellative is a necessary condition for a semigroup to be embeddable in a group.
Note that a non-empty semigroup is a group if and only if it is cancellative and regular.
$S$ is weakly cancellative if, for all $a,b,c\in S$ $(ab=ac~\&~ba=ca)\Rightarrow b=c$ A semigroup is completely simple if and only if it is weakly cancellative and regular.
An element $x\in S$ is called left cancellative if, for all $b,c\in S$ $xb=xc\Rightarrow b=c$ An element $x\in S$ is called right cancellative if, for all $b,c\in S$ $bx=cx\Rightarrow b=c$
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"cancellative semigroup" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: cancellation ideal
| Other names: |
cancellation semigroup |
| Also defines: |
cancellative, weakly cancellative, left cancellative, right cancellative, weakly cancellative semigroup, left cancellative semigroup, right cancellative semigroup |
This object's parent.
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Cross-references: completely simple, regular, necessary, groups, semigroup
There are 4 references to this entry.
This is version 6 of cancellative semigroup, born on 2004-06-17, modified 2006-10-04.
Object id is 5926, canonical name is CancellativeSemigroup.
Accessed 7719 times total.
Classification:
| AMS MSC: | 20M10 (Group theory and generalizations :: Semigroups :: General structure theory) |
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Pending Errata and Addenda
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