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Inmathematics, the notion ofcancellativity (orcancellability) is a generalization of the notion ofinvertibility that does not rely on an inverse element.
An elementa in amagma(M, ∗) has theleft cancellation property (or isleft-cancellative) if for allb andc inM,a ∗b =a ∗c always implies thatb =c.
An elementa in a magma(M, ∗) has theright cancellation property (or isright-cancellative) if for allb andc inM,b ∗a =c ∗a always implies thatb =c.
An elementa in a magma(M, ∗) has thetwo-sided cancellation property (or iscancellative) if it is both left- and right-cancellative.
A magma(M, ∗) is left-cancellative if alla in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties.
In asemigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. Ifa−1 is the left inverse ofa, thena ∗b =a ∗c impliesa−1 ∗ (a ∗b) =a−1 ∗ (a ∗c), which impliesb =c by associativity.
For example, everyquasigroup, and thus everygroup, is cancellative.
To say that an element in a magma(M, ∗) is left-cancellative, is to say that the functiong :x ↦a ∗x isinjective wherex is also an element ofM.[1] That the functiong is injective implies that given some equality of the forma ∗x =b, where the only unknown isx, there is only one possible value ofx satisfying the equality. More precisely, we are able to define some functionf, the inverse ofg, such that for allx,f(g(x)) =f(a ∗x) =x. Put another way, for allx andy inM, ifa ∗x =a ∗y, thenx =y.[2]
Similarly, to say that the elementa is right-cancellative, is to say that the functionh :x ↦x ∗a is injective and that for allx andy inM, ifx ∗a =y ∗a, thenx =y.
The positive (equally non-negative) integers form a cancellativesemigroup under addition. The non-negative integers form a cancellativemonoid 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 (as the above examples clearly do) will obey the cancellative law.
In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of aring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is adomain, like the integers) has the cancellation property. This remains valid even if the ring in question is noncommutative and/or nonunital.
Although the cancellation property holds for addition and subtraction ofintegers,real andcomplex numbers, it does not hold for multiplication due to exception of multiplication byzero. The cancellation property does not hold for any nontrivial structure that has anabsorbing element (such as 0).
Whereas the integers and real numbers are not cancellative under multiplication, with the removal of 0, they each form a cancellative structure under multiplication.