Several Studies Of Flat Semirings

Flat semirings is an important class of additively idempotent semirings.It plays an important role in the study of semiring varieties.In this dissertation,the basic properties of flat semirings and the varieties generated by a special class of flat nil-semirings are studied.the main results are as follows:1.We determine the relationship between flat semirings and 0-cancellative semigroups,and obtain the congruences on the flat semirings are exactly the Ress congruences,which therefore one-to-one corresponds to the ideal of multiplicative reduct of semirings.2.We characterize the basic properties of the flat nil-semirings,and give the necessary and sufficient conditions for the subdirectly irreducibility of flat nil-semiring.3.We introduce the definition of 0-ω direct union of flat semirings,and characterize the structure of the subdirectly irreducible members in V（S（x₁x₂···x_n））.It is determined that the flat semirings S（x₁x₂···x_n）are finitely based when n=1,2,3.