Numerical Semigroups, Polyhedra, and Posets III: Minimal Presentations and Face Dimension

This paper is the third in a series of manuscripts that examine combinatorics Kunz polyhedron $P_m$, whose positive integer points are bijection with numerical semigroups (cofinite subsemigroups $\mathbb Z_{\ge 0}$) smallest element $m$. The faces $P_m$ indexed by family finite posets (called posets) obtained from divisibility lying on given face. In this paper, we characterize to what extent minimal presentation semigroup can be recovered its poset. doing so, prove all interior face have identical cardinality, and provide combinatorial method obtaining dimension corresponding