Positioned and primary positioned $\mathcal{C}$-semigroups

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 $k\in \mathcal{C}$.