On Some Studies Of Semirings

In this thesis,we mainly discussed the congruence extension property for semirings and the homomorphism theorem for ordered semirings.Particularly,we investigated the congruence extension property for a special rectangular ring.Also,we discussed the subdirect product of ordered semirings.We divided the thesis into three chapters.In chapter 1,we introduced the historical background of the semiring theory and its important application value in mathematics.We gave some basic concepts and results which would used in the following chapters.In chapter 2,we mainly investigated the congruence extension property for semirings.At first,we provided the definition of bi-ideal semirings and biideal extension property of a semiring.Then we gave some examples that a bi-ideal semiring which has the bi-ideal extension property does not have the congruence extension property,a subsemiring of a bi-ideal semiring may not be a bi-ideal semiring,etc.Finally,a necessary and sufficient condition of a special rectangular ring with congruence extension property was established.In chapter 3,homomorphism theorems for ordered semirings are studied.We obtained some characterizations of the smallest order-congruence or compatible quasiorder generated by H.A homomorphism theorem and two isomorphism theorems of ordered semirings based on the order-congruence and compatible quasiorder have been given.Finally,we observed the relation between compatible quasiorder and subdirect product of ordered semirings.