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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Monomorphism ： ウィキペディア英語版 Monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from ''X'' to ''Y'' is often denoted with the notation $X\; \backslash hookrightarrow\; Y$. In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism, that is, an arrow such that, for all morphisms , : $f\; \backslash circ\; g\_1\; =\; f\; \backslash circ\; g\_2\; \backslash Rightarrow\; g\_1\; =\; g\_2.$ Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, i.e. a monomorphism in a category ''C'' is an epimorphism in the dual category ''C''op. Every section is a monomorphism, and every retraction is an epimorphism. ==Relation to invertibility== Left invertible morphisms are necessarily monic: if ''l'' is a left inverse for ''f'' (meaning ''l'' is a morphism and $l\; \backslash circ\; f\; =\; \backslash operatorname\_$), then ''f'' is monic, as : $f\; \backslash circ\; g\_1\; =\; f\; \backslash circ\; g\_2\; \backslash Rightarrow\; lfg\_1\; =\; lfg\_2\; \backslash Rightarrow\; g\_1\; =\; g\_2.$ A left invertible morphism is called a split mono. However, a monomorphism need not be left-invertible. For example, in the category Group of all groups and group morphisms among them, if ''H'' is a subgroup of ''G'' then the inclusion is always a monomorphism; but ''f'' has a left inverse in the category if and only if ''H'' has a normal complement in ''G''. A morphism is monic if and only if the induced map , defined by for all morphisms , is injective for all ''Z''. 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「Monomorphism」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.