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

TL;DR

This paper studies a special class of semigroups called positioned and primary positioned $\\mathcal{C}$-semigroups, providing characterizations, properties, and computational methods for these algebraic structures within positive integer cones.

Contribution

It introduces primary positioned $\\mathcal{C}$-semigroups, characterizes them via irreducibility, and develops algorithms and graph-based methods to compute all such semigroups.

Findings

Characterization of primary positioned $\\mathcal{C}$-semigroups.

Development of procedures to compute these semigroups.

Description of a graph family containing all primary positioned $\\mathcal{C}$-semigroups.

Abstract

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 $k\in \mathcal{C}$.