Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

Motivated by the construction of Steenrod cup-i products in singular cochain algebra a space and non-commutative differential forms, we define category binomial cup-one graded algebras over integers prime fields positive characteristic. The Hirsch identities bind cup-product, product, package that further enhance with ring structure arising from integer-valued rational polynomials. This allows us to free generated set derive its basic properties. It also provides context for defining restricted triple Massey products, which have smaller indeterminacy than classical ones, hence, give stronger homotopy type invariants.