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Generalizations of the series exp and log to noncommutative nonassociative and other types of algebras were considered by M. Lazard, and recently by V. Drensky and L. Gerritzen. There is a unique power series exp(x) in one non-associative variable x such that exp(x) exp(x) = exp(2x), exp ′ (0) = 1.We call the unique series H = H(x, y) in two non-associative variables satisfying exp(H) = exp(x) exp(y) the non-associative Hausdorff series, and we show that the homogeneous components H n of H are primitive elements with respect to the coaddition for non-associative variables. We describe the space of primitive elements for the co-addition in non-associative variables using Taylor expansion and a projector onto the algebra A 0 of constants for the partial derivations. By a theorem of Kurosh, A 0 is a free algebra. We describe a procedure to construct a free algebra basis consisting of primitive elements.

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Formal group schemes, associated to affine group schemes or Lie groups by completion, can be described by classical formal group laws. More generally, Ž . cogroup objects in categories of complete algebras e.g., associative are described by group laws for operads or analyzers. M. Lazard has introduced analyzers to Ž . study formal group laws and group law chunks truncated formal power series . A main example of a type of generalized formal group laws not given by an operad or analyzer are group laws corresponding to noncommutative complete Hopf algebras. To cover this case and other types of group laws, pseudo-analyzers are Ž . introduced. We point out differences to the quadratic operad case; e.g., there is no classification of group laws by Koszul duality. On the other hand we show how pseudo-analyzer cohomology can be used to describe extension of group law chunks.