Some Congruences On Several Classes Of Semirings

Congruence and ideal are major tools in studying the structure of semiring. Many scholars have deeply, systematically studied on them. As we well known, the relation between the ideal and congruence of ring is one to one. The skew-ring congruence of regular semiring and qusi-regular semiring are characterized by using of a class of special ideals of semirings and a construction method of regulal subdirect product of regular semirings is given by using of a special mapping on semirings is also studied in this paper.The paper consists of five chapters. In Chapter 1, the background and the present state of the semiring theory and some necessary fundamental knowledge about semigroup and semiring are simply introduced. In Chapter 2, properties of group congruences on the regular semigroups are briefly introduced. In Chapter 3, the skew-ring congruence of regular semiring and qusi-regular semiring are characterized by using of a class of special ideals and a representation of a class of a skew-ring congruence is given by using of a quasi-order on semirings. In Chapter 4, a congruence on a commutative distributive semiring is defined by using of a style of congruence in reference [16]. Moreover, the result that the congruence is the least distributive lattice congruence is proved. In Chapter 5, a construction method of regular subdirect product of regular semirings is given by using of a special mapping on semiring.