Some Studies About Semirings

In this thesis, some kinds of congruences on semirings or inverse semirings arestudied. The thesis is divided into four chapters, and the last three chapters arethree integrated papers.In Chapter 1, the background of semiring theory is introduced brie?y. Also,some basic concepts and theorems about semigroup and semiring are listed.In Chapter 2, the maximum additive idempotent-separating congruenceμandminimum skew-ring congruenceσon an inverse semiring S are described. Thecongruenceσ∨μgenerated by the two congruences is studied.In Chapter 3, we investigate the relationship between ring congruences and nor-mal subsemirings . As an application of this relation, if the right ring congruencesof a semiring S exist, the structure of S is determined and the right ring congru-ences are characterized in terms of ring congruences and right zero band semiringcongruences on S.In Chapter 4, the relationship between distributive lattice of inverse semiringsand distributive lattice of skew-rings Gαis described. This relationship is used tostudy distributive lattices of E+-unitary inverse semirings. In particular, it is shownthat if S is a distributive lattice of E+-unitary inverse semirings, then there existsa minimum distributive lattice congruenceηsuch that eachη-class is E+-unitaryinverse semiring and there exists a maximum distributive lattice congruenceνsuchthat eachν-class is an E+-unitary inverse semiring.