On Isolated Gaps of Non-perfect Arf Numerical Semigroups

This study examines the structure of non-perfect Arf numerical semigroups and explores the characteristics of their isolated gaps. The analysis focuses on the structural relationship between the elements of Arf numerical semigroups and their gap sets. For non-perfect Arf numerical semigroups, the position and behavior of isolated gaps, together with their connection to the Frobenius number, are characterized. It is shown that the union of an Arf numerical semigroup with its set of isolated gaps forms a perfect Arf numerical semigroup. Moreover, when certain subsets of isolated gaps are considered, the resulting semigroups preserve their non-perfect nature. These findings shed new light on the relationship between perfect and non-perfect Arf numerical semigroups and clarify the role of isolated gaps in the transition between these two classes.