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

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.