Conference abstracts

Session S05 - Rings and Algebras

No date set

## Construction of $\text{Rota}^{m}\text{-Algebras}$ and $\text{Ballot}^{m}\text{-Algebras}$ from Associative Algebras with a Rota-Baxter morphism and a Rota-Baxter Operator of Weights Three and Two

### Wilson Arley Martinez

### Universidad del Cauca, Colombia - wamartinez@unicauca.edu.co

We give a generalization of Rota-Baxter Operators and introduce the notion of a Ballot$^{m}$-algebra. Free Rota-Baxter algebras on a set can be constructed from a subset of planar rooted forests with decorations on the angles. We give similar constructions for obtaining an associative algebra in terms of planar binary trees with a modified Rota-Baxter Operator, and so we construct a Ballot$^{m}$-algebra.

We introduce the concepts of a Rota-Baxter Morphism, Dyck$^{m}$-algebra and Rota$^{m}$-algebra. An element $u$ is said to be idempotent with respect to product $\cdot$ in the algebra if: $u \,\cdot\, u = u,$ and it is a left identity if $x \,\cdot\, u = x$ for all element $x$ in the algebra. Associative algebras with a left identity that simultaneously is a element idempotent, permit us to present examples of a Rota-Baxter Morphism and so we can construct a Rota$^{m}$-algebra.

We stress that the construction of Ballot$^{m}$-algebras and Rota$^{m}$-algebras from associative algebras with a generalitation of Rota-Baxter Operators are some of the main results of this work.