Cancellative element: Difference between revisions
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==Definition== | ==Definition== | ||
An element <math>a</math> in a set <math>S</math> with binary operation <math>*</math> is termed: | An element <math>a</math> in a [[magma]] <math>(S,*)</math> (a set <math>S</math> with binary operation <math>*</math>) is termed: | ||
* '''left cancellative''' if whenever <math>a * b = a * c</math>, <math>b = c</math> | * '''left-cancellative''' if whenever <math>a * b = a * c</math>, <math>b = c</math> | ||
* '''right cancellative''' if whenever <math>b * a = c * a</math>, <math>b = c</math> | * '''right-cancellative''' if whenever <math>b * a = c * a</math>, <math>b = c</math> | ||
* '''cancellative''' if it is both left and right cancellative | * '''cancellative''' if it is both left and right cancellative | ||
A magma where every element is left-cancellative (resp. right-cancellative, cancellative) is termed a [[left-cancellative magma]] (resp., [[right-cancellative magma]], [[cancellative magma]]). | |||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 20:50, 31 July 2008
This article defines a property of elements or tuples of elements with respect to a binary operation
Definition
An element in a magma (a set with binary operation ) is termed:
- left-cancellative if whenever ,
- right-cancellative if whenever ,
- cancellative if it is both left and right cancellative
A magma where every element is left-cancellative (resp. right-cancellative, cancellative) is termed a left-cancellative magma (resp., right-cancellative magma, cancellative magma).
Relation with other properties
Stronger properties
- Invertible element: In a monoid, any left invertible element is right cancellative, any right invertible element is left cancellative. Thus, any invertible element is cancellative. For full proof, refer: invertible implies cancellative in monoid