Reynolds代数及其括号词和有根树中的自由对象
Reynolds algebras and their free objects from bracketed words and rooted trees
摘要：
Reynolds代数的研究起源于O。雷诺兹在1895年发表了关于流体动力学的论文，并从此得到了广泛的应用。它与代数自同态、导子、Rota-Baxter算子等重要的线性算子也有着密切的关系。多年前，G伯克霍夫建议对雷诺算子进行代数研究，包括相应的自由代数。我们在本文中进行了这样的研究。我们首先提供的例子和雷诺算子的属性，包括多变量的雷诺恒等式的推广。然后我们在一个集合上构造自由Reynolds代数。为了这个目的，我们确定了一组括号中的话，称为雷诺的话，作为线性基础的自由雷诺代数。Reynolds词的一个组合的解释是根据有根树没有超级冠。Reynolds词在连接下的闭包给出了Reynolds词所张空间上的代数结构。然后在此代数上定义一个线性算子，使得雷诺恒等式和所需的泛性质得到满足。
英文摘要：
The study of Reynolds algebras has its origin in the well-known work of O. Reynolds on fluid dynamics in 1895 and has since found broad applications. It also has close relationship with important linear operators such as algebra endomorphisms, derivations and Rota-Baxter operators. Many years ago G.~Birkhoff suggested an algebraic study of Reynolds operators, including the corresponding free algebras. We carry out such a study in this paper. We first provide examples and properties of Reynolds operators, including a multi-variant generalization of the Reynolds identity. We then construct the free Reynolds algebra on a set. For this purpose, we identify a set of bracketed words called Reynolds words which serves as the linear basis of the free Reynolds algebra. A combinatorial interpretation of Reynolds words is given in terms of rooted trees without super crowns. The closure of the Reynolds words under concatenation gives the algebra structure on the space spanned by Reynolds words. Then a linear operator is defined on this algebra such that the Reynolds identity and the desired universal property are satisfied.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
