On Some Additive Properties of Multiplicative Subsemigroups of Semirings and Arithmetic Applications I

In this paper, we consider a question of sum-keeping about a multiplicative subsemigroup and its generator subsets in a semiring, and develop some elementary (collapse) process of the sum-keeping retraction through subsets until one minimal generators subset. As an application, we study and analyze several classical problems in additive number theory on the semiring of non-negative integers by this algebraic and combinatory idea, and provide new proofs in more simple and direct way for several classical results in number theory. Some further questions are also presented and discussed.