Typed Angularly Decorated Planar Rooted Trees and Ω-Rota-Baxter Algebras

As a generalization of Rota–Baxter algebras, the concept of an Ω-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an Ω-dendriform algebra and show the relationship between Ω-Rota–Baxter algebras and Ω-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free Ω-Rota–Baxter algebra by typed, angularly decorated rooted trees.

Matching BiHom-Rota-Baxter Algebras and Related Structures

In this paper, we introduce the notions of matching BiHom-Rota-Baxter algebras, matching BiHom-(tri)dendriform algebras, matching BiHom-Zinbiel algebras and matching BiHom-pre-Lie algebras. Moreover, we study the properties and relationships between categories of these matching BiHom-algebraic structures.

Generalizing dendriform algebras: Dyckm-algebras, Rotam-algebras, and Rota–Baxter operators

We review the notion of a [Formula: see text], an algebraic structure introduced recently by López, Préville-Ratelle and Ronco during their work on the splitting of associativity via [Formula: see text]-Dyck paths, and we also introduce Rota[Formula: see text]-algebras: both structures can be considered as generalizations of dendriform structures. We obtain examples of Dyck[Formula: see text]-algebras in terms of planar rooted binary trees equipped with a particular type of Rota–Baxter operator, and we present examples of Rotam-algebras using left averaging morphisms. As an application, we observe that the structures presented here allow us to introduce quite naturally a “non-associative version” of the Kadomtsev–Petviashvili hierarchy.

3-L-dendriform algebras and generalized derivations

The main goal of this paper is to introduce the notion of 3-L-dendriform algebras which are the dendriform version of 3-pre-Lie algebras. In fact they are the algebraic structures behind the O-operator of 3-pre-Lie algebras. They can be also regarded as the ternary analogous of L-dendriform algebras. Moreover, we study the generalized derivations of 3-L-dendriform algebras. Finally, we explore the spaces of quasi-derivations, the centroids and the quasi-centroids and give some properties.