On finitely generated submonoids of \(\mathbb{N}^ k\) (Q1346309)

A commutative additively written monoid is said to be cancellative if \(a+b=a+c\) implies \(b=c\); it is torsion free if for any positive integer \(k\) the equality \(ka=kb\) implies \(a=b\) and is called reduced if \(u+v=0\) implies \(u=v=0\). It is shown that a commutative and finitely generated monoid is for some \(k \geq 1\) isomorphic to a submonoid of \(N^k\) if and only if it is cancellative, torsion-free and reduced. Moreover an algorithm is given to check whether these conditions are satisfied for monoids of the form \(N^n/M\) with \(M\) being a subgroup of \(N^n\). Another algorithm is also presented which may be used to check whether a homogeneous system of linear equations with rational coefficients has a non-trivial solution in natural numbers.