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In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the right cancellation property (or is right-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative. A magma has the left cancellation property (or 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. A left-invertible element is left-cancellative, and analogously for right and two-sided. For example, every quasigroup, and thus every group, is cancellative. ==Interpretation== To say that an element ''a'' in a magma is left-cancellative, is to say that the function is injective, so a set monomorphism but as it is a set endomorphism it is a set section, i.e. there is a set epimorphism ''f'' such for all ''x'', so ''f'' is a retraction. Moreover, we can be "constructive" with ''f'' taking the inverse in the range of ''g'' and sending the rest precisely to ''a''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cancellation property」の詳細全文を読む スポンサード リンク
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