A combinatorial problem and numerical semigroups

Abstract

Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two $n$-tuples of positive integers, let $X$ be a set of positive integers, and let $g$ be a positive integer. In this work we show an algorithmic process in order to compute all the sets $C$ of positive integers that fulfill the following conditions: 1) the cardinality of $C$ is equal to $g$; 2) if $x,y\in \mathbb{N} \setminus \{0\}$ and $x+y\in C$, then $C \cap \{x,y\} \neq \emptyset$; 3) if $x \in C$ and $\frac{x-b_i}{a_i} \in \mathbb{N} \setminus \{0\}$ for some $i\in \{1,\ldots,n\}$, then $\frac{x-b_i}{a_i} \in C$; 4) $X \cap C = \emptyset$.