Irreducible Numerical Semigroup With Multiplicity Five And Six

A numerical semigroup S is a subset of N(here N denotes the set of non-negative integers)closed under addition,containing zero and such that N\ S is finite.Numerical semigroups were first considered while studying the set of nonnegative solutions of Diophantine equations and their study is closely related to the analysis of monomial curves [1].For these reasons,the theory of digital semigroups has attracted many researchers in the field of algebra and geometry.Base on the irreducible digital semigroups with the weights of 3 and 4,this paper combines the existing results and generalizes them to study the irreducible digital semigroups with the number of 5 and 6.The details are as follows:In the first chapter,the background and significance of this study are introduced fir stly,and then the research progress and main conclusions of this paper are introduced.In the second chapter,we briefly introduce the development direction of digital semigroups and introduce some basic knowledge of digital semigroups related to this paper.In the third chapter,the definitions of irreducible digital semigroups and some related conclusions are given,and the irreducibility characteristics of irreducible digital semigroups with weights of 3 and 4 are given.The fourth chapter is the main part of this thesis.First,we give some basic conclusions of the digital semigroup with a weight of 5.Secondly,based on the simple conclusion,we pass the irreducible digital semigroup with a multiplicity of 5.The relationship between the irreducible digital semigroups,which concludes that,except{0,5,6,8,?},{0,5,?},{0,5,7,?},any irreducible number semigroup must be irreducible.This conclusion is then demonstrated in the form of a list.Finally,by the combination of the proof and the list,the relationship between the irreducible number semigroup with the number of 6 and the irreducible number semigroup is obtained.These results have certain application value for studying the relationship between irreducible digital semigroups and irreducible digital semigroups.