Editing Cancellation property (section)

{{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.