Closed factors of normal \(\mathbb Z\)-semimodules (Q1054782)

Let \(M\) be a subset of the positive integers such that \(x,y\in M\) implies \(xy\in M\), and if \(y\) divides \(x\), then \(x/y\in M\). A closed factor of \(M\) is a subset \(K\) of \(M\) such that \(x,y\in K\) implies \(xy\in K\) and there exists a subset \(R\) of \(M\) such that every element \(m\in M\) has a unique representation \(m=kr\) with \(k\in K\), \(r\in R\). A theory is developed for determining all closed factors of \(M\). This work makes a substantial contribution to the factorization problems surveyed in the author's introduction. In general, we are given a semigroup \(S\) say, and wish to find two subsets \(K, R\) of \(S\) such that every element \(s\) of \(S\) has a unique factorization \(s=kr\) with \(k\in K\), \(r\in R\). The origin of such problems and their subsequent treatment is detailed by the author, together with the references which we will not repeat here. An adaption of the theory is applied to an analogous problem for convex polyhedral cones.