Typed angularly decorated planar rooted trees and generalized Rota-Baxter algebras

TL;DR

This paper generalizes parametrized Rota-Baxter algebras by introducing structures on parameter sets, describing free algebras via decorated trees, and defining new algebraic notions like $mbda$-extended diassociative semigroups.

Contribution

It introduces a broad generalization of Rota-Baxter algebras, including new structures on parameters and combinatorial descriptions of free algebras.

Findings

Defined $mbda$-extended diassociative semigroups.

Described free Rota-Baxter algebras using decorated planar rooted trees.

Provided explicit constructions for free commutative $mbda$-Rota-Baxter algebras.

Abstract

We introduce a generalization of parametrized Rota-Baxter algebras, which includes family and matching Rota-Baxter algebras. We study the structure needed on the set $\Omega$ of parameters in order to obtain that free Rota-Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of $\lambda$-extended diassociative semigroup, which includes sets (for matching Rota-Baxter algebras) and semigroups (for family Rota-Baxter algebras), and many other examples. We also describe free commutative $\Omega$-Rota-Baxter algebras generated by a commutative algebra A in terms of typed words.