Rota--Baxter and averaging operators on racks and rack algebras

TL;DR

This paper introduces and explores relative Rota--Baxter and averaging operators on racks and rack algebras, establishing their properties and connections to algebraic operators, with implications for extending these operators linearly.

Contribution

It defines and investigates relative Rota--Baxter and averaging operators on racks and rack algebras, and connects these to operators on arbitrary algebras, including linear extensions.

Findings

Extension of averaging operators from racks to rack algebras via linearity.

Connections established between Rota--Baxter and averaging operators.

Linear extension preserves averaging operator properties.

Abstract

In the present article we define and investigate relative Rota--Baxter operators and relative averaging operators on racks and rack algebras. Also, if B is a Rota--Baxter or averaging operator on a rack X, then we can extend B by linearity to the rack algebra k[X]. On the other side, we have definitions of Rota--Baxter and averaging operators on arbitrary algebra. We find connections between these operators. In particular, we prove that if B : X --> X is an averaging operator on a rack, then its linear extension on a rack algebra k[X] gives an averaging operator.