Editing Cancellation property

{{Short description|Extension of "invertibility" in abstract algebra}}
{{About|the extension of 'invertibility' in [[abstract algebra]]|cancellation of terms in an [[equation]] or in [[elementary algebra]]|cancelling out}}
{{More references|date=December 2009}}
In [[mathematics]], the notion of '''cancellativity''' (or ''cancellability'') is a generalization of the notion of [[invertibility]].

An element ''a'' in a [[magma (algebra)|magma]] {{nowrap|(''M'', ∗)}} has the '''left cancellation property''' (or is '''left-cancellative''') if for all ''b'' and ''c'' in ''M'', {{nowrap|1=''a'' ∗ ''b'' = ''a'' ∗ ''c''}} always implies that {{nowrap|1=''b'' = ''c''}}.

An element ''a'' in a magma {{nowrap|(''M'', ∗)}} has the '''right cancellation property''' (or is '''right-cancellative''') if for all ''b'' and ''c'' in ''M'', {{nowrap|1=''b'' ∗ ''a'' = ''c'' ∗ ''a''}} always implies that {{nowrap|1=''b'' = ''c''}}.

An element ''a'' in a magma {{nowrap|1=(''M'', ∗)}} has the '''two-sided cancellation property''' (or is '''cancellative''') if it is both left- and right-cancellative.

A magma {{nowrap|(''M'', ∗)}} is left-cancellative if  all ''a'' 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 ''a''<sup>−1</sup> is the left inverse of ''a'', then {{nowrap|1=''a'' ∗ ''b'' = ''a'' ∗ ''c''}} implies {{nowrap|1=''a''<sup>−1</sup> ∗ (''a'' ∗ ''b'') = ''a''<sup>−1</sup> ∗ (''a'' ∗ ''c'')}}, which implies {{nowrap|1=''b'' = ''c''}} by associativity.

For example, every [[quasigroup]], and thus every [[group (mathematics)|group]], is cancellative.

== Interpretation ==
To say that an element ''a'' in a magma {{math|(''M'', ∗)}} is left-cancellative, is to say that the function {{math|''g'' : ''x'' ↦ ''a'' ∗ ''x''}} is [[injective]].<ref>{{cite book |last1=Warner |first1=Seth |title=Modern Algebra Volume I |date=1965 |publisher=Prentice-Hall, Inc. |location=Englewood Cliffs, NJ |page=50}}</ref> That the function ''g'' is injective implies that given some equality of the form {{math|1=''a'' ∗ ''x'' = ''b''}}, where the only unknown is ''x'', there is only one possible value of ''x'' satisfying the equality. More precisely, we are able to define some function ''f'', the inverse of ''g'', such that for all ''x'' {{math|1=''f''(''g''(''x'')) = ''f''(''a'' ∗ ''x'') = ''x''}}. Put another way, for all ''x'' and ''y'' in ''M'', if {{math|1=''a'' * ''x'' = ''a'' * ''y''}}, then {{math|1=''x'' = ''y''}}.<ref>{{cite book |last1=Warner |first1=Seth |title=Modern Algebra Volume I |date=1965 |publisher=Prentice-Hall, Inc. |location=Englewood Cliffs, NJ |page=48}}</ref>

Similarly, to say that the element ''a'' is right-cancellative, is to say that the function {{math|''h'' : ''x'' ↦ ''x'' ∗ ''a''}} is injective and that for all ''x'' and ''y'' in ''M'', if {{math|1=''x'' ∗ ''a'' = ''y'' ∗ ''a''}}, then {{math|1=''x'' = ''y''}}.

== Examples of cancellative monoids and semigroups ==
The positive (equally non-negative) 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 (as the above examples clearly do) will obey the cancellative law.

In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a [[Ring (mathematics)|ring]] that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a [[Domain (ring theory)|domain]], like the integers) 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 law holds for addition, subtraction, multiplication and division of [[real number|real]] and [[complex number]]s (with the single exception of multiplication by [[0 (number)|zero]] and division of zero by another number), there are a number of algebraic structures where the cancellation law is not valid.

<!-- The [[vector (spatial)|vector]] [[dot product]] is perhaps the simplest example. In this case, for an arbitrary nonzero vector '''a''', the product {{nowrap|1='''a''' ⋅ '''b'''}} can equal another dot product {{nowrap|1='''a''' ⋅ '''c'''}} even if {{nowrap|'''b''' ≠ '''c'''}}. This occurs because the dot product relates to the angle between two vectors as well as their magnitude, and a change in one can, in effect, counterbalance the other to produce equal products for unequal vectors.

For the same reason, the-->The [[cross product]] of two vectors <!--also--> does not obey the cancellation law. If {{nowrap|1='''a''' × '''b''' = '''a''' × '''c'''}}, then it does not follow that {{nowrap|1='''b''' = '''c'''}} even if {{nowrap|'''a''' ≠ '''0'''}} (take {{nowrap|1='''c''' = '''b''' + '''a'''}} for example)

<!-- However, if ''both'' '''a''' · '''b'''='''a''' · '''c''' ''and'' '''a''' × '''b''' = '''a''' × '''c''', then one ''can'' conclude that '''b''' = '''c'''. This is because for dot and cross products to be simultaneously equal, then both '''a''' · ('''b''' - '''c''') ''and'' '''a''' × ('''b''' - '''c''') must be zero by the [[distributive law]]. This means that both the sine and cosine of the angle between '''a''' and ('''b''' - '''c''') must be zero, which is not possible because sin<sup>2</sup>''x'' + cos<sup>2</sup>''x'' is ''identically'' 1.-->
[[Matrix multiplication]] also does not necessarily obey the cancellation law. If {{nowrap|1='''AB''' = '''AC'''}} and {{nowrap|'''A''' ≠ 0}}, then one must show that matrix '''A''' is ''invertible'' (i.e. has {{nowrap|[[determinant|det]]('''A''') ≠ 0}}) before one can conclude that {{nowrap|1='''B''' = '''C'''}}. If {{nowrap|1=det('''A''') = 0}}, then '''B''' might not equal '''C''', because the [[matrix (mathematics)|matrix]] equation {{nowrap|1='''AX''' = '''B'''}} will not have a unique solution for a non-invertible matrix '''A'''.

Also note that if {{nowrap|1='''AB''' = '''CA'''}} and {{nowrap|'''A''' ≠ 0}} and the matrix '''A''' is ''invertible'' (i.e. has {{nowrap|[[determinant|det]]('''A''') ≠ 0}}), it is not necessarily true that {{nowrap|1='''B''' = '''C'''}}. Cancellation works only for {{nowrap|1='''AB''' = '''AC'''}} and {{nowrap|1='''BA''' = '''CA'''}} (provided that matrix '''A''' is ''invertible'') and not for {{nowrap|1='''AB''' = '''CA'''}} and {{nowrap|1='''BA''' = '''AC'''}}.

== See also ==
* [[Grothendieck group]]
* [[Invertible element]]
* [[Cancellative semigroup]]
* [[Integral domain]]

== References ==
{{reflist}}

{{DEFAULTSORT:Cancellation Property}}
[[Category:Non-associative algebra]]
[[Category:Properties of binary operations]]
[[Category:Algebraic properties of elements]]

[[fr:Loi de composition interne#Réguliers et dérivés]]