Cancellation property

Kürzbarkeit is a property of the elements of an algebraic structure.

• 2.1 Definition
• 2.2 Examples

Shortenable / regular items

Given a groupoid / Magma.

Definition

An element is called linkskürzbar or left regular if applies to all:

And rechtskürzbar or right- regular if for all cases:

Ie two length can be shortened or two sides regularly or just shortened or regular if c is left and rechtskürzbar.

Remark

* Is commutative, however, all three types of Kürzbarkeit are the same are generally not.

Example

• In a ring is an element if and only be shortened if it is a non- zero divisor.

Shortenable / regular semigroups

Definition

A semigroup is called shortened or regular if each one is shortened.

Examples

• The set of natural numbers with the usual addition and with the usual multiplication is a semigroup can be shortened.
• The set of natural numbers with the maximum or the minimum is not reducible semigroup.

• Algebra