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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 initial algebra ： ウィキペディア英語版 initial algebra In mathematics, an initial algebra is an initial object in the category of ''F''-algebras for a given endofunctor ''F''. The initiality provides a general framework for induction and recursion. For instance, consider the endofunctor 1+(-) on the category of sets, where 1 is the one-point set, the terminal object in the category. An algebra for this endofunctor is a set ''X'' (called the ''carrier'' of the algebra) together with a point and a function . The set of natural numbers is the carrier of the initial such algebra: the point is zero and the function is the successor map. For a second example, consider the endofunctor 1+N×(-) on the category of sets, where N is the set of natural numbers. An algebra for this endofunctor is a set ''X'' together with a point and a function . The set of finite lists of natural numbers is the initial such algebra. The point is the empty list, and the function is cons, taking a number and a finite list, and returning a new finite list with the number at the head. ==Final coalgebra== Dually, a final coalgebra is a terminal object in the category of ''F''-coalgebras. The finality provides a general framework for coinduction and corecursion. For example, using the same functor 1+(-) as before, a coalgebra is a set $X$ together with a truth-valued test function $p\backslash colon\; X\; \backslash to\; 2$ and a partial function $f\backslash colon\; X\; \backslash to\; X$ whose domain is formed by those $x\; \backslash in\; X$ for which $p(x)\; =\; 0$. The set $\backslash mathbb\; \backslash cup\; \backslash$ consisting of the natural numbers extended with a new element $\backslash omega$ is the carrier of the final coalgebra in the category, where $p$ is the test for zero: $p(0)=1$ and $p(n+1)\; =\; p(\backslash omega)\; =\; 0$, and $f$ is the predecessor function (the inverse of the successor function) on the positive naturals, but acts like the identity on the new element $\backslash omega$: $f(n+1)\; =\; n$, $f(\backslash omega)\; =\; \backslash omega$. For a second example, consider the same functor $1\; +\; \backslash mathbb\backslash times(\backslash mathord)$ as before. In this case the carrier of the final coalgebra consists of all lists of natural numbers, finite as well as infinite. The operations are a test function testing whether a list is empty, and a deconstruction function defined on nonempty lists returning a pair consisting of the head and the tail of the input list. 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「initial algebra」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.