Type sequences of numerical semigroups

Numerical semigroups have a very important place in Algebra and Number Theory. Numerical semigroups, which were first discussed in the 19th century, have become the focus of attention over time. With the researches, new definitions have emerged and many studies have been carried out in this field until today. In recent years, we have seen that extensive applications of numerical semigroups have been included in the fields of Topology, Algebraic Geometry and Differential Geometry. Particularly the applications in Group and Ring theory are remarkable. Studies on numerical semigroups are not limited to these. In this thesis, it is aimed to compile the previous studies on numerical semigroups and their type sequences, to compare the results obtained and to express the new findings as a result of this. In this direction, general information about numerical semigroups and their type sequences, basic definitions, important theorems and propositions compiled from various sources are included in our thesis. The concepts of Frobenius number, Pseudo Frobenius number, Symmetry, Pseudo symmetrical and Arf numerical semigroup concepts, which have an important place in the theory of numerical semigroups, are examined with examples. In addition, the relation of type sequences with symmetric and pseudo-symmetric numerical semigroups and type sequences of numerical semigroups with determine numbers 2, 3 and 4 are given. In our thesis, besides the different methods that can be used to find the type sequence of a numerical semigroup, necessary and sufficient conditions for numerical semigroups corresponding to a given positive integer sequence are given. At the same time, some algorithms where we can find the type sequences of numerical semigroups, different from these methods, are mentioned