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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 Rational series ： ウィキペディア英語版 Rational series In mathematics and computer science, a rational series is a generalisation of the concept of formal power series over a ring to the case when the basic algebraic structure is no longer a ring but a semiring, and the indeterminates adjoined are not assumed to commute. They can be regarded as algebraic expressions of a formal language over a finite alphabet. ==Definition== Let ''R'' be a semiring and ''A'' a finite alphabet. A ''noncommutative polynomial'' over ''A'' is a finite formal sum of words over ''A''. They form a semiring $R\backslash langle\; A\; \backslash rangle$. A ''formal series'' is a ''R''-valued function ''c'', on the free monoid ''A'' *, which may be written as :$\backslash sum\_\; c(w)\; w\; \backslash \; .$ The set of formal series is denoted $R\backslash langle\backslash langle\; A\; \backslash rangle\backslash rangle$ and becomes a semiring under the operations :$c+d\; :\; w\; \backslash mapsto\; c(w)\; +\; d(w)\; \backslash$ :$c\backslash cdot\; d\; :\; w\; \backslash mapsto\; \backslash sum\_\; c(u)\; \backslash cdot\; d(v)\; \backslash \; .$ A non-commutative polynomial thus corresponds to a function ''c'' on ''A'' * of finite support. In the case when ''R'' is a ring, then this is the ''Magnus ring'' over ''R''. If ''L'' is a language over ''A'', regarded as a subset of ''A'' * we can form the ''characteristic series'' of ''L'' as the formal series :$\backslash sum\_\; w\; \backslash$ corresponding to the characteristic function of ''L''. In $R\backslash langle\backslash langle\; A\; \backslash rangle\backslash rangle$ one can define an operation of iteration expressed as :$S^$ * = \sum_ S^n \ and formalised as :$c^$ *(w) = \sum_ c(u_1)c(u_2) \cdots c(u_n) \ . The ''rational operations'' are the addition and multiplication of formal series, together with iteration. A rational series is a formal series obtained by rational operations from $R\backslash langle\; A\; \backslash rangle$. 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「Rational series」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.