Can be shortened

Shortenability is a property of elements of an algebraic structure .

Can be shortened / regular elements

Given is a groupoid / magma . ${\displaystyle (M,*)}$

definition

An element is called left- shortable or left-regular if the following applies to all : ${\displaystyle c\in M}$ ${\displaystyle a,b\in M}$

${\displaystyle c*a=c*b\implies a=b,}$

and legally abbreviated or legally regular , if the following applies to all : ${\displaystyle a,b\in M}$

${\displaystyle a*c=b*c\implies a=b.}$

${\displaystyle c\in M}$means can be shortened on both sides or regular on both sides or simply can be shortened or regular if it can be shortened to the left and right. ${\displaystyle c}$

comment

If * is commutative , all three types of shortening are the same, but generally not.

example

• An element in a ring can be shortened if it is a non-zero divisor . ${\displaystyle (R,+,\cdot )}$
• In a quasi-group , all elements can be shortened.

Shortened / regular half-groups

definition

A semigroup is called shortenable or regular if each can be shortened. ${\displaystyle (S,*)}$ ${\displaystyle a\in S}$

Examples

• The set of natural numbers with the usual addition or with the usual multiplication is a semigroup that can be shortened. ${\displaystyle (\mathbb {N} ,+)}$ ${\displaystyle (\mathbb {N} ,\cdot )}$
• The set of natural numbers with the maximum or with the minimum is not a semigroup that can be reduced. ${\displaystyle (\mathbb {N} ,{\text{max}})}$ ${\displaystyle (\mathbb {N} ,{\text{min}})}$