Abstract
We introduce a generalization of parametrized Rota–Baxter algebras, named \(\Omega \)-Rota–Baxter algebra, which includes family and matching Rota–Baxter algebras. We study the structure needed on the set \(\Omega \) of parameters in order to obtain that free \(\Omega \)-Rota–Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of \(\lambda \)-extended diassociative semigroup, which includes sets (for matching Rota–Baxter algebras) and semigroups (for family Rota–Baxter algebras), and many other examples. We also describe free commutative \(\Omega \)-Rota–Baxter algebras generated by a commutative algebra A in terms of typed words.
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