A partial magmatic bialgebra, or (T; S)-magmatic bialgebra, where T ⊂ S are subsets of N ≥ 2, is a vector space endowed with an n-ary operation for each n ∈ S and an m-ary co-operation for each m ∈ T satisfying some compatibility and unitary relations. We prove an analogue of the Poincaré-Birkhoff-Witt theorem for these partial magmatic bialgebras.
Emily Burgunder. Ralf Holtkamp. "Partial magmatic bialgebras." Homology Homotopy Appl. 10 (2) 59 - 81, 2008.