We extend past results on a family of formal power series K n,Λ , parameterized by n and Λ ⊆ [n], that largely resemble quasisymmetric functions. This family of functions was conjectured to have the property that the product K n,Λ K m,Ω of any two functions K n,Λ and K m,Ω from the family can be expressed as a linear combination of other functions from the family. In this paper, we show that this is indeed the case and that the span of the K n,Λ 's forms an algebra. We also provide techniques for examining similar families of functions and a formula for the product K n,Λ K m,Ω when n = 1.