Presentations of a numerical semigroups

ABSTRACT PRESENTATIONS OF A NUMERICAL SEMIGROUPS KANBAY, Sibel M.Sc. in Mathematics Supervisor: Assoc. Prof. Dr. Belgin ÖZER June 2020 44 pages The object of this paper is to study the presentations of finitely generated numerical semigroups. The study of presentations of numerical semigroups are mainly motivated by its applications to Algebraic Geometry (see [7], [8]). Redei in [9] shows that every congruence on N^n is finitely generated. This result is since then known as Redei's Theorem. Many other authors have given alternative and quite much simpler proofs than his ([10,11,12,13]). Since numerical semigroups are cancellative monoids , a different approach can be chosen to prove Redei's theorem. In this paper we focus on the computation of a (and in fact all) minimal presentation of a numerical semigroup. The idea comes from Rosales' PhD Thesis ([15]) and was published later in [16]. The cardinality of a numerical semigroup cannot be bounded in terms of its embedding dimension. This follows from Bresinsky's family of embedding dimension four numerical semigroups, which have arbitrarily large minimal presentations ([17]). In this paper we offer an upper bound for the cardinality of a minimal presentation in terms of the multiplicity of the semigroup. Moreover we examine the concept of gluing, the cardinality , the connectedness, the complete intersection, the catenary degree,elasticity of some numerical semigroups. Key Words: Minimal Presentations of Numerical Semigroups, Gluing, Complete Intersection, Catenary Degree, Elasticity.