© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2026.Let be a positive integer cone and. A -semigroup S is k-positioned if for every we have that belongs to S. In this work, we focus on this family of semigroups and introduce primary positioned -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 -semigroups for a fixed.