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We extend past results on a family of formal power series $K_{n, \Lambda}$,parameterized by $n$ and $\Lambda \subseteq [n]$, that largely resemblequasisymmetric functions. This family of functions was conjectured to have theproperty that the product $K_{n, \Lambda}K_{m, \Omega}$ of any two functions$K_{n, \Lambda}$ and $K_{m, \Omega}$ from the family can be expressed as alinear combination of other functions from the family. In this paper, we showthat this is indeed the case and that the span of the $K_{n, \Lambda}$'s formsan algebra. We also provide techniques for examining similar families offunctions and a formula for the product $K_{n, \Lambda}K_{m, \Omega}$ when$n=1$.

On Quasisymmetric Functions with Two Bordering Variables