Numerical semigroups

In this study, by compiling the studies related with numerical semigroups so far, aimed to explain the new proof which will be gained by comparing the results between this studies. In the first section of the thesis, with the literature summary of numerical semigroups the hypothesis has been stated by producing the purpose in the preparation of the thesis. In the II. section of the thesis, it is given information like that in Euclid, GCD-LCM and Diophant equations which is thought provided on the basic issues for the numerical semigroups. In the III. section of the thesis, it is mentioned the basic definition of numerical semigroups like that the minimal generating set, the Frobenius number, multiplicity, the set of gaps and the Pseudo-Frobenius numbers. In the IV. section of the thesis, the Apery sets and the properties of the Apery sets are stated and the set providing the Theorem 4.3 under which conditions it may be an Apery set has been explained and proved. In the V. section of the thesis, symmetric, pseudo-symmetric and almost symmetric numerical semigroups are introduced and the transitions between them are exemplified. In the VI. section of the thesis, it is mentioned about the set of fundamental gaps and the determination of the numerical semigroup that has given fundamental gaps and is specified the conditions for being a numerical semigroups if an element is added or removed. In the last section, it is mentioned the ideals of numerical semigroups and the types of the ideals are explained.