Cancellative element
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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