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In mathematics, specifically in category theory, ''F''-algebras generalize algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor ''F'', the ''signature''. ''F''-algebras can also be used to represent data structures used in programming, such as lists and trees. The main related concepts are initial ''F''-algebras which may serve to encapsulate the induction principle, and the dual construction ''F''-coalgebras. ==Definition== If ''C'' is a category, and ''F'': ''C'' → ''C'' is an endofunctor of ''C'', then an ''F''-algebra is a tuple (''A'', α), where ''A'' is an object of ''C'' and α is a ''C''-morphism ''F''(''A'') → ''A''. The object ''A'' is called ''carrier'' of the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple. A homomorphism from an ''F''-algebra (''A'', α) to an ''F''-algebra (''B'', β) is a ''C''-morphism ''f'': ''A''→ ''B'' such that ''f'' o α = β o ''F''(''f''), according to the following diagram: center Equipped with these morphisms, ''F''-algebras constitute a category. The dual construction are ''F''-coalgebras, which are objects ''A'' * together with a morphism α * : ''A'' * → ''F''(''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「F-algebra」の詳細全文を読む スポンサード リンク
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