Research On Erd(?)s-Burgess Constant On Semigroups

The Erd(?)s-Burgess constant of semigroups is an invariant in the zero-sum theory.Zero-sum theory is one research branch of Combinatorial Number Theory.The Erd(?)s-Burgess constant of a semigroup S is defined as the smallest l ?N?{?} such that any sequence of terms from S and of length at least l contains a nonempty subsequence the product of whose terms,in some order,is an idempotent of the semigroup S.The Erd(?)s-Burgess constant was originated from one question of Erd(?)s proposed to Burgess,in which was conjectured that every sequence of terms from any finite semigroup S of order n must contain some terms whose product in some order is idempotent.Moreover,The Erd(?)s-Burgess constant reduces to be the classical Davenport constant when the semigroup S happens to be a group.In this thesis,the Erd(?)s-Burgess constant is studied on the multiplicative semigroup of the ring of integers modulo n and on the the multiplicative semigroup of any quotient rings of a polynomial ring over finite field Fq.In Chapter one,we introduce the background of the thesis,and necessary notations,preliminaries and related theorems,and we also describe how we arrange the sections in the thesis.In Chapter two,we introduce the obtained results on the Erd(?)s-Burgess constant in the past,and we illustrate the relationship between the Erd(?)s-Burgess constant and the Davenport constant.In Chapter three,we study the Erd(?)s-Burgess constant on the multiplicative semigroup of the quotient ring of the polynomial ring Fq[x]and of the ring Zn of integers modulo n.For the quotient ring Fq[x]/K,we give a sharp lower bound of the Erd(?)s-Burgess constant of its multiplicative semigroup and determine the Erd(?)s-Burgess constant in the case when K is the power of a prime ideal or a product of pairwise distinct prime ideals.For Zn,we give a sharp lower bound of the Erd(?)s-Burgess constant of its multiplicative semigroup and determine the Erd(?)s-Burgess constant in the case when n is a prime power or a product of pairwise distinct primes.For the multiplicative semigroup of a full matrix ring over finite field F2,we give a lower bound of the Erd(?)s-Burgess constant.In Chapter four,we provide an overview of recent work by Kravitz and Sah,who based on our results,have shown that our conjecture on the Erd(?)s-Burgess constant holds true when n=2pk,n=pkql,n=2pkql.In addition,they give an upper bound on the Erd(?)s-Burgess constant when n=spk,and give the Erd(?)s-Burgess constant a lower bound of the multiplicative semigroup of the quotient ring of the unique factorized domains and the Dedekind domain.