A variation of gluing of numerical semigroups

Abstract

For a numerical semigroup \(S=\left \), we consider a numerical semigroup which is in the form of \(T=\left \), where \(d>1\) and \(\gcd (d, a_n)=1\). When \(a_n \in \left \) 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 \(\left \) and \(\mathbb {N}\) in Rosales and Garcia-Sanchez (Numerical semigroups, 2009). In this paper, we study the relation between S and T in the case where \(a_n \notin \left \).