Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

The mod 2 Steenrod algebra A and Dyer-Lashof algebra R have both striking s imilarities and differences arising from their common origins in "lower-ind exed" algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, bu t quite different from, those of A and R. The exact relationships emerge as "sheared algebra bijections", which also illuminate the role of the cohomo logy of K. As a bialgebra, K* has a particularly attractive and potentially useful structure, providing a bridge between those of A* and R*, and sugge sting possible applications to the Miller spectral sequence and the A struc ture of Dickson algebras.