The Frobenius problem over number fields with a real embedding

Given a number field K with at least one real embedding, we generalize the notion of the classical Frobenius problem to the ring of integers \({\mathfrak {O}}_K\) of K by describing certain Frobenius semigroups, \(\textrm{Frob}(\alpha _1,\ldots ,\alpha _n)\), for appropriate elements \(\alpha _1,\ldots ,\alpha _n\in {\mathfrak {O}}_K\). We construct a partial ordering on \(\textrm{Frob}(\alpha _1,\ldots ,\alpha _n)\), and show that this set is completely described by the maximal elements with respect to this ordering. We also show that \(\textrm{Frob}(\alpha _1,\ldots ,\alpha _n)\) will always have finitely many such maximal elements, but in general, the number of maximal elements can grow without bound as n is fixed and \(\alpha _1,\ldots ,\alpha _n\in {\mathfrak {O}}_K\) vary. Explicit examples of the Frobenius semigroups are also calculated for certain cases in real quadratic number fields.