Let \(\mathcal {C}\) be a positive integer cone and \(k\in \mathcal {C}\) . A \(\mathcal {C}\) -semigroup S is k-positioned if for every \(h\in \mathcal {C}\setminus S\) we have that \(k-h\) belongs to S. In this work, we focus on this family of semigroups and introduce primary positioned \(\mathcal {C}\) -semigroups, characterizing a subfamily of them through the perspective of irreducibility. Furthermore, we provide some procedures to compute all such semigroups, describing a family of graphs containing all the primary positioned \(\mathcal {C}\) -semigroups for a fixed \(k\in \mathcal {C}\) .
