Compositions of a numerical semigroup

Given a numerical semigroup $S$, a nonnegative integer $a$ and $m\in S\backslash\{0\}$, we introduce the set $C(S,a,m)=\{s+aw(s~mod~m)~|~s\in S\}$, where $\{w(0), w(1), \cdots, w(m-1)\}$ is the Apéry set of $m$ in $S$. In this paper we characterize the pairs $(a,m)$ such that $C(S,a,m)$ is a numerical semigroup. We study the principal invariants of $C(S,a,m)$ which are given explicitly in terms of invariants of $S$. We also characterize the compositions $C(S,a,m)$ that are symmetric, pseudo-symmetric and almost symmetric. Finally, a result about compliance to Wilf's conjecture of $C(S,a,m)$ is given.