Classification of coproducts compatible with rooted-tree shuffles

Classify all coproducts that turn the rooted-tree shuffle algebra $(\mathcal{T}_\Omega,\shuffle^T_\lambda)$ into a bialgebra, including the possibility that coproducts other than the one constructed in the paper exist.

Background

The paper constructs a coproduct that is compatible with the nonassociative shuffle product of rooted trees and thereby obtains a bialgebra structure. The authors indicate that computations suggest the existence of additional coproducts with the same compatibility property. Determining whether such coproducts exist and classifying all of them is identified as a subject for future studies.

References

Computation suggest there are might be other coproducts that turn $(\calT_\Omega,\shuffleT_\lambda)$ into a bialgebra. A classification is beyond the scope of this work and left for future studies.

— Coalgebras, bialgebras and Rota-Baxter algebras from shuffles of rooted forests  (2501.02557 - Clavier et al., 5 Jan 2025) in Remark following Theorem 1.?, Section 1, "Shuffle bialgebra"