Generalized numerical semigroups and special classes

If S ⊆ N_0^d is a monoid and N_0^d\S is finite, S is known as a generalized numerical semigroup. Generalized numerical semigroups provide a more general framework for the Frobenuis problem, which is important in classical numerical semigroup theory, and contain more complex structures than numerical semigroups. Therefore, generalized numerical semigroups provide important analysis and discovery. This thesis focuses on introducing various classes of generalized numerical semigroups, investigating their properties, determining and analyzing the relationships between these classes. Regarding these new structures, generalized perfect numerical semigroups, generalized sparse, dense, fully dense numerical semigroups, and the arithmetic extensions of the generalized numerical semigroup are defined. Properties such as the Frobenius element, Apery set, generator systems, and genus have been examined, and interesting examples have been provided for each class. It will contribute to the classical numerical semigroup families by introducing some properties of triangular and pyramidal numerical semigroups in N0, especially the minimal generator systems, and examining the n-th β-Pascal and θ-gonal numerical semigroups, which are two generalizations of triangular numerical semigroups.