Reynolds algebras and their free objects from bracketed words and rooted trees

TL;DR

This paper develops a combinatorial and algebraic framework for Reynolds algebras, constructing free Reynolds algebras using bracketed words and rooted trees, and explores their properties and operators.

Contribution

It introduces Reynolds words as a basis for free Reynolds algebras and provides a combinatorial interpretation using rooted trees, advancing the algebraic understanding of Reynolds operators.

Findings

Defined Reynolds words and their properties.

Constructed free Reynolds algebras with a basis of Reynolds words.

Provided a combinatorial interpretation via rooted trees.

Abstract

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.