In this paper by Nondas Kechagias, the Steenrod algebra and the Dyer–Lashof algebra are compared. The rough difference ist:

- The Steenrod algebra arises by dividing out the “cohomological” Ádem relations and $Q^0=1$.
- The Dyer–Lashof algebra arises by dividing out the “homological” Ádem relations and $Q^I=0$ where $I$ has negative excess.

The cohomological Ádem relations are of the form $$Q^iQ^j =\sum_{k=0}^{[i/2]}\binom{j-k-1}{i-2k}Q^{i+j-k}Q^k~~~\text{for }i<2j$$ The homological Ádem relations are of the form $$Q^iQ^j-\sum_{k>0}\binom{k-j-1}{2k-i}Q^{i+j-k}Q^k~~~\text{for }i>2j.$$ In what way are they dual to each other? If I consider the dual Steenrod sqares $\mathrm{Sq}_i:H_*(X)\to H_{*-i}(X)$ for $X$ of finite type, the should satisfy other Adem relations than the “homological ones”, namely the cohomological ones with the only difference that the composition has to be read contravariantly.

Why do we call both relations Ádem relations?