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 .${\ displaystyle (R, +, \ cdot)}$

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}})}$