Infinite magmatic bialgebras

An infinite magmatic bialgebra is a vector space endowed with n-ary operations, and n-ary cooperations, for each n, verifying some compatibility relations. We prove an analogue of the Hopf–Borel theorem for infinite magmatic bialgebras. We show that any connected infinite magmatic bialgebra is of the form Mag∞(PrimH), where Mag∞(V) is the free infinite magmatic algebra over the vector space V.