Convex polytopes and factorization properties in generalized power series domains
An element \(a\) of an integral domain \(R\) is primal if, whenever \(a\) divides \(bc\) with \(b\) and \(c\) in \(R\), then \(a=b'c'\) for some \(b',c'\in R\) where \(b'\) divides \(b\) and \(c'\) divides \(c\). An integral domain in which each element is primal is said to be pre-Schreier. Let \((M,\leq)\) be a strictly ordered monoid; that is, \(M\) is a commutative monoid and \(\leq\) is a partial order on \(M\) such that \(x<y\) implies \(x+z<y+z\) for all \(x,y,z\in M\). A subset \(N\) of \(M\) is said to be narrow if each subset of \(N\) consisting of pairwise order-incomparable elements in the \(\leq\) order is finite. Let \(R\) be a commutative ring. For a function \(f:M\longrightarrow R\) the support of \(f\) is defined as \(\mathrm{supp}(f)=\{x\in M; f(x)\not=0\}\). Then the generalized power series ring \(RM,\leq \) is the set of all such functions whose support is Artinian and narrow in the \(\leq\) partial ordering. Addition is defined by \((f+g)(x)=f(x)+g(x)\) and multiplication by \((fg)(x)=\sum_{x_1+x_2=x}f(x_1)g(x_2)\). Let \(K\) be any commutative field. In this paper the authors associate to any polytope \(C\subseteq R^n\), that is not simplex, a monoid \(M={M}_s(C)\) such that the domain \(R=KM,\leq\), of generalized power series, has no irreducible elements and that is not pre-Schreier.
scientific article; zbMATH DE number 5541302