Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
A family Rota-Baxter is an algebra equipped with a family of operators \((R_\alpha)_{\alpha\in \Omega}\) indexed by a fixed semigroup \(\Omega\), such that for any \(\alpha,\beta\in \Omega\), the following axiom is satisfied: \[ R_\alpha(x)R_\beta(y)=R_{\alpha\beta}(R_\alpha(x)y+xR_\beta(y)+\lambda ab), \] where \(\lambda\) is a scalar. This paper generalizes Guo's construction of free Rota-Baxter algebras based on planar leaf-spaced rooted forests. This generalization used planar leaf-spaces rooted trees which leaves are decorated by the generators and edges are typed (or decorated) by elements of \(\Omega\), with constraints on the types. The Rota-Baxter operators just add a root (and a typed edge) and the product is inductively defined using the family Rota-Baxter axiom.
scientific article; zbMATH DE number 7435542
scientific article