On the elasticity of a numerical semigroup

Let $S$ be a numerical semigroup, and $\operatorname{msg}(S)$ its minimal system of generators. Then $\mathrm{m}(S)=\min(\operatorname{msg}(S))$, $\mathrm{M}(S)=\max(\operatorname{msg}(S))$, $\mathrm{e}(S)$, which is the cardinality of $\operatorname{msg}(S)$, and $\mathscr{S}(S)=\frac{\mathrm{M}(S)}{\mathrm{m}(S)}$, are called the multiplicity, comultiplicity, embedding dimension, and elasticity of $S$, respectively. Let $m$ and $M$ be positive integers, and let $q$ be a rational number greater than $1$. In this paper, we will study the following sets: \begin{itemize} \item $\{S\mid S \mbox{ is a numerical semigroup}, \mathrm{m}(S)=m \mbox{ and } \mathrm{M}(S)=M\}$, \item $\{S\mid S \mbox{ is a numerical semigroup}, \mathrm{m}(S)=m \mbox{ and } \mathscr{S}(S)\leq q\}$, and \item $\{S\mid S \mbox{ is a numerical semigroup}, \mathrm{e}(S)=3 \mbox{ and } \mathscr{S}(S)=q\}$. \end{itemize}