Decorated planar rooted forests carry lambda-parameter bialgebra coproducts satisfying a symmetric 1-cocycle condition, and at lambda equals zero they form the free Omega-cocycle Hopf algebra on the leaf labels.
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Moerdijk Hopf algebras of decorated rooted forests: an operated algebra approach
Decorated planar rooted forests carry lambda-parameter bialgebra coproducts satisfying a symmetric 1-cocycle condition, and at lambda equals zero they form the free Omega-cocycle Hopf algebra on the leaf labels.