Cancellation property
In mathematics, the notion of cancellativity is a generalization of the notion of invertibility that does not rely on an inverse element.
An element in a magma has the left cancellation property if for all and in , always implies that .
An element in a magma has the right cancellation property if for all and in , always implies that .
An element in a magma has the two-sided cancellation property if it is both left- and right-cancellative.
A magma is left-cancellative if all in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties.
In a semigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. If is the left inverse of , then implies, which implies by associativity.
For example, every quasigroup, and thus every group, is cancellative.
Interpretation
To say that an element in a magma is left-cancellative, is to say that the function is injective where is also an element of. That the function is injective implies that given some equality of the form, where the only unknown is , there is only one possible value of satisfying the equality. More precisely, we are able to define some function , the inverse of , such that for all ,. Put another way, for all and y in , if, then .Similarly, to say that the element is right-cancellative, is to say that the function is injective and that for all and y in , if, then .
Examples of cancellative monoids and semigroups
The positive integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. Each of these is an example of a cancellative magma that is not a quasigroup.Any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid that embeds into a group will obey the cancellative law.
In a different vein, the multiplicative semigroup of elements of a ring that are not zero divisors has the cancellation property. This remains valid even if the ring in question is noncommutative and/or nonunital.
Non-cancellative algebraic structures
Although the cancellation property holds for addition and subtraction of integers, real and complex numbers, it does not hold for multiplication due to exception of multiplication by zero. The cancellation property does not hold for any nontrivial structure that has an absorbing element.Whereas the integers and real numbers are not cancellative under multiplication, with the removal of, they each form a cancellative structure under multiplication.