A variation of gluing of numerical semigroups

Abstract

For a numerical semigroup $S=\left<a_1,...,a_n\right>$, we consider a numerical semigroup which is in the form of $T=\left<da_1,...,d a_{n-1}, a_n\right>$, where $d>1$ and $\gcd (d, a_n)=1$. When $a_n \in \left<a_1,...,a_{n-1}\right>$ and $a_n \ne a_i$ for any $1 \le i \le n-1$, the numerical semigroups of this form were studied in Watanabe (Nagoya Math J 49:101–109, 1973), and T is called a gluing of $\left<a_1,...,a_{n-1}\right>$ and $\mathbb {N}$ in Rosales and García-Sánchez (Numerical semigroups, 2009). In this paper, we study the relation between S and T in the case where $a_n \notin \left<a_1,...,a_{n-1}\right>$.