  In this paper we examine the descriptive potential of a combinatorial data
structure known as "Generating Set" in constructing the boundary maps of a
simplicial complex. By refining the approach of \cite{Dumas} in generating
these maps, we provide algorithms that allow for relations among simplices to
be easily accounted for. In this way we explicitly generate faces of a complex
only once, even if a face is shared among multiple simplices. The result is a
useful interface for constructing complexes with many relations and for
extending our algorithms to $\Delta$-complexes. Once we efficiently retrieve
the representatives of "living" simplices i.e., of those that have not been
related away, the construction of the boundary maps scales well with the number
of relations and provides a simpler alternative to JPlex. We finish by noting
that the generating data of a complex is equivalent in information to its
incidence matrix and we provide efficient algorithms for converting from an
incidence matrix to a Generating Set.
