ON THE DEFINITION OF PI-ALGEBRAS

A II-algebra is a graded group {pi(n)}(n greater than or equal to 1) together with an action of the primary homotopy operations. Using James-Hopf invariants and certain words analogous to the classical Hall-Petrosco words, a set of relations is described which generate all universal relations of these operations. Moreover, a category J is described such that the II-algebras are precisely the internal groups in J.