Coassociative magmatic bialgebras and the Fine numbers. (Q1024806)

A magmatic algebra is a vector space equipped with a unital binary operation. A magmatic bialgebra is a magmatic algebra which is additionally equipped with a counital binary cooperation satisfying some natural compatibility axioms. In the paper under review the authors prove a structure theorem for connected coassociative magmatic bialgebras. The authors show that the space of primitive elements of such a bialgebra is an algebra over an operad, which gives rise to an equivalence of categories between the category of connected coassociative magmatic bialgebras and a certain category of magmatic algebras. In particular, it follows that any connected coassociative magmatic bialgebra is cofree. This statement is analogous to the PBW theorem for (classical) cocommutative bialgebras. Fine numbers appear as dimensions of the space of \(n\)-ary operations of the primitive operad mentioned above.