### A Cellular Construction of BP and Other Irreducible Spectra.

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The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.

We classify the indecomposable injective E(n)⁎E(n)-comodules, where E(n) is the Johnson-Wilson homology theory. They are suspensions of the ${J}_{n,r}=E\left(n\right)\u204e\left({M}_{r}E\left(r\right)\right)$, where 0 ≤ r ≤ n, with the endomorphism ring of ${J}_{n,r}$ being $\widehat{E\left(r\right)}*\widehat{E\left(r\right)}$, where $\widehat{E\left(r\right)}$ denotes the completion of E(r).

A family of multiplicative operations in the BP Steenrod algebra is defined which is related to the total Steenrod power operation from the mod p Steenrod algebra. The main result of the paper links the BP versions of the total Steenrod power with the formal group approach to multiplicative BP operations by identifying the p-typical curves (power series) which correspond to these operations. Some relations are derived from this identification, and a short proof of the Hopf invariant one theorem...