A combinatorial problem and numerical semigroups

Abstract

Let $a=(a_1,\dots ,a_n)$ and $b=(b_1,\dots ,b_n)$ 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 $C\cap \{x,y\}\ne \varnothing$; 3) if $x\in C$ and $\frac{x-b_i}{a_i}\in \mathbb{N}\setminus \{0\}$ for some $i\in \{1,\dots ,n\}$, then $\frac{x-b_i}{a_i}\in C$; 4) $X\cap C=\varnothing$.