Numerical semigroups with quasi maximal embedding dimension

Consider \(x\in {\mathbb {N}}\setminus \{0\}\). A QMED(x)-semigroup is a numerical semigroup S such that \(S{\setminus }\{0\}=\{a+kx \mid a\in {\text {msg}}(S) \text{ and } k\in {\mathbb {N}}\}\) where \({\text {msg}}(S)\) denotes the minimal system of generators of S. Note that if x is the multiplicity of S then S is a maximal embedding dimension numerical semigroup. In this work, we show that the set of all QMED(x)-semigroups is a Frobenius pseudo-variety giving the associated tree. Furthermore, we give formulas to obtain the Frobenius number, type, and genus of this class of semigroups.