Let \(\text{mon}(A)= \{Ax\mid x\in \mathbb{Z}_+^n\}\), where \(A\in \mathbb{Z}^{m\times n}\). An element \(g\in \text{mon}(A)\) is called a swelling point if \((g+ \text{cone}(A))\cap \mathbb{Z}^m \subseteq \text{mon}(A)\), where \(\text{cone}(A)= \{Ax\mid x\in \mathbb{R}_+^n\}\). Then the main result is the following: Let \(C\) be the set of all swelling points of \(\text{mon}(A)\). Then there exists a finite subset \(T\subset C\) such that every \(x\in C\) has a representation \(x=y+ A\lambda\) with \(y\in T\) and \(\lambda\in \mathbb{Z}_+^n\).
If there exists a finite set \(M\subset C\) such that \((M+ \text{cone}(A)) \cap \mathbb{Z}^m= C\) and no proper subset of \(M\) has this property, then we call \(M\) the minimal set of \(C\). The set \(T\) then contains the minimal set \(M\) of \(C\). There are connections to the generalized Frobenius problem.