This paper deals with the problem of recognizability of functions l: Sigma* --> M that map words to values in the support set M of a monoid (M,.,1). These functions are called M-languages. M-languages are studied from the aspect of their recognition by deterministic finite automata whose components take values on M (M-DFAs). The characterization of an M-language l is based on providing a right congruence on Sigma* that is defined through l and a factorization on the set of all M-languages, L(Sigma*,M) (in sort L). A factorization on L is a pair of functions (g,f) such that, for each l in L, g(l). f(l)= l, where g(l) in M and f(l) in L. In essence, a factorization is a form of common factor extraction. A general Myhill-Nerode theorem, which is valid for any L(Sigma*, M), is provided. Basically, l is recognized by an M-DFA if and only if there exists a factorization on L, (g,f), such that the right congruence on Sigma* induced by the factorization (g,f) and f(l), has finite index. This paper shows that the existence of M-DFAs guarantees the existence of natural non-trivial factorizations on L without taking account any additional property on the monoid.

中文翻译：

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通过具确定性的有限自动机对语言进行可识别性，其确定值在一个半群上：一般的Myhill-Nerode定理

本文讨论了函数l的可识别性问题：Sigma *-> M，该函数将单词映射到一个等分体（M，。，1）的支持集M中的值。这些功能称为M语言。从确定性有限自动机的角度对M语言进行了研究，确定性自动机的成分取M值（M-DFA）。M语言l的表征基于对通过l定义的Sigma *提供正确的一致性，以及对所有M语言L（Sigma *，M）的集合（在L类中）进行因式分解。对L的因式分解是一对函数（g，f），这样，对于L中的每个l，g（l）。f（l）= l，其中M中的g（l）和L中的f（l）。本质上，因式分解是公因子提取的一种形式。提供了适用于任何L（Sigma *，M）的通用Myhill-Nerode定理。基本上，当且仅当在L（g，f）上存在因式分解时，M DFA才识别l，使得因因式分解（g，f）和f（l）引起的Sigma *的正确同余性是有限的指数。本文表明，M-DFA的存在保证了L上自然非平凡因式分解的存在，而没有考虑到类半群上的任何其他性质。