Shortenability is a property of elements of an algebraic structure .
Can be shortened / regular elements
Given is a groupoid / magma .
![(M, *)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b49d2845cb12b058528061e86914ae5b99ca1da3)
definition
An element is called left- shortable or left-regular if the following applies to all :
![c \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d253c4736a68b7e0cb89c5a256494c0de78ea2)
![a, b \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb00e3fe8849131890d2b4db8373457573bd0a4)
![c * a = c * b \ implies a = b,](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fabac0b4a34e1b707cc50f86223b6d8b95e822d)
and legally abbreviated or legally regular , if the following applies to all :
![a, b \ in M](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb00e3fe8849131890d2b4db8373457573bd0a4)
![a * c = b * c \ implies a = b.](https://wikimedia.org/api/rest_v1/media/math/render/svg/03f2250e6e74f6320262a8468d7ff7c6d1491b08)
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.
![c](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
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 .
![(R, +, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
- 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.
![(S, *)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d4d31503b4a9a01b9115a35d2a6d89d5eebd9e)
![a \ in S](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe43ae205afd3f10432bddabc3bdae5ecfa6b412)
Examples
- The set of natural numbers with the usual addition or with the usual multiplication is a semigroup that can be shortened.
![({\ mathbb N}, +)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0072a6ee0ab943ce24dc44083bd60d50739a0b1f)
![({\ mathbb N}, \ cdot)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7217221cfdf4bf46efa7dbf99a7e0f53195baa1a)
- The set of natural numbers with the maximum or with the minimum is not a semigroup that can be reduced.
![({\ mathbb N}, {\ text {min}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8d40265c2cc4ad9157abd1273e76bd2e4ae03e)