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[parent] cancellative semigroup (Definition)

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.

Relationship to some other types of semigroup

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.

Individual elements

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$



"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

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Cross-references: completely simple, regular, necessary, groups, semigroup
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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 4850 times total.

Classification:
AMS MSC: 20M10 (Group theory and generalizations :: Semigroups :: General structure theory)
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