On the algebra of operations for Hopf cohomology

In his thesis (mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations Sq(k) on the cohomology ring of any cocommutative Hopf algebra over Z/2. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that Sq(0) is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra B that is generated by elements Sqk and subject to Adem relations; it shows that the Cartan formula gives a well-defined coproduct on B. Also, it is shown that B with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name 'algebra with coproducts' is proposed in the paper.