一种广义的无限词幺半群：渐近前后缀商与代数结构
A generalized monoid of infinite words: asymptotic prefix-suffix quotients and algebraic structure
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中文总结 AI 辅助
本研究构造了基于双向前后缀度量渐近等价的广义词幺半群\\(\tilde{\Sigma}^*\\)，揭示其代数性质，证明它严格扩张经典无限词集合，并建立了对数前缀泛函的良定义作用。
中文摘要 AI 辅助
我们构造了广义词幺半群\\(\tilde{\Sigma}^*\\)，它是有限词的适度网（moderate nets）在基于双向前后缀度量的渐近等价关系下的商结构。主要代数结果表明：该商结构是一个忠实包含\\(\Sigma^*\\)的幺半群，具备反转对合、自然可除预序，且不满足消去律、存在非平凡幂等元。该构造的设计使得任何仅依赖对数前缀的泛函都可下降到商结构上，从而得到良定义的对数前缀泛函作用。有限标量值仅在施加额外的重整化泛函后才会出现，该泛函依赖于所选的mould和窗口。\n相对于固定的截断单射，该幺半群严格扩张了经典集合\\(\Sigma^{\infty}\\)：我们构造了一个显式的振荡网，它在Cantor空间中没有极限，但定义了\\(\tilde{\Sigma}^*\\)中的一个真实元素，且在该单射下并非来自有限词或右无限词。
英文摘要
A generalized monoid of words tilde{Sigma}* is constructed as the quotient of moderate nets of finite words by an asymptotic equivalence relation based on a bidirectional prefix-suffix metric. The main algebraic result is that this quotient is a monoid containing Sigma* faithfully, with a reversal involution, a natural divisibility preorder, failure of cancellation, and nontrivial idempotents. The construction is designed so that any functional depending only on a logarithmic prefix descends to the quotient, yielding a well-defined action of logarithmic-prefix functionals. Finite scalar values arise only after applying an additional renormalization functional, which depends on the chosen mould and window. With respect to the fixed truncation injection, the monoid strictly enlarges the classical set Sigma^{infty}: an explicit oscillating net is exhibited that has no limit in the Cantor space but defines a genuine element of tilde{Sigma}* not coming from a finite or right-infinite word under that injection.
