A unitary infinitesimal bialgebra of weight zero is a quadruple 
in which 
is a unital associative algebra, 
is a coassociative coproduct,
and
 |
for all 
.
Thus the coproduct obeys a derivation
rule with respect to the product. A counit is not required.
Yu and Gao (2026) construct such a structure on the module spanned by planar binary trees, using the under product and a root-recursive coproduct. They prove that it is isomorphic to a known infinitesimal bialgebra on planar rooted forests and is distinct from the usual Loday-Ronco Hopf algebra structure.
