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