Algebraic construction of Steenrod operations

Steenrod squares $\mathrm{Sq}^k$ and Steenrod powers $P^k$, the generators of mod $2$ and mod $p$ Steenrod algebras, are kind of linear cohomology operations. Construction of these operations requires Eilenberg–MacLane spaces, $\smile_i$-products, acyclic carriers, etc. But their algebraic constructions are more convenient and realizable. L. Smith did such construction in 2007 and used them to solve some invariant theoretic problems.
In this talk, we will speak on algebraic construction of Steenrod operations and comprehensive algebraic constructions of these operations. We will construct a family of $R$-algebras $\left\{ \Sigma_n(R)\right\}_{n\geq0}$ for $R$ a commutative ring with identity, with special cases $\Sigma_2(\mathbb{F}_2)$ being mod $2$ Steenrod algebra and $\Sigma_{p^n}(\mathbb{F}_{p^n})$, the algebras of L. Smith. We use this algebras to solve linear differential equations.