On Some Non Perfect Numerical Semigroups With Multiplicity Three

In this study, we investigate numerical semigroups of multiplicity three of the form Wq= , focusing on their structural properties within the class of maximal embedding dimension (MED) semigroups. The aim is to determine the position of this class with respect to perfect, Arf, and symmetric numerical semigroups and to describe the behavior of their fundamental invariants. Our results show that these semigroups are non-perfect, possess the Arf property, and are not symmetric. Moreover, the Frobenius number, isolated gaps, Apery set, and pseudo-Frobenius numbers exhibit a regular structure within this class. The findings contribute to the structural classification of MED and Arf numerical semigroups and provide potential applications in the study of value semigroups, algebraic curve singularities, and coding theory.