Some Studies On Semirings From The Angle Of Semigroups

The study of the theory of semirings is one of the important topics in algebra. Many experts and scholars have carried out thorough,painstaking and systematical research into it.Semiring may be regarded as the two semigroups related by the distributive laws on the same nonempty set.Thus,we can study the semirings by means of the viewpoints and methods in the algebraic theory of semigroups.We mainly study the structures of semiring with a completely regular multiplicative(additive)semigroup from the angle of semigroups in this paper.The dissertation consists of six chapters.In chapter one,the background,the present state of semiring theory and some fundamental knowledge about semigroups and semirings are simply introduced.In chapter two,the subdirect product decompositions of the A(M)-completely regular semirings are studied by virtue of the theory of the sturdy frame of type(2,2)algebras from the angle of the additive(multiplicative) semigroup and some results of paper[27]are extended.In chapter three,firstly,many characterizations of the semirings in the Mal'cev product(?)I are given by using the Green's(?)-relation on a semiring;secondly,the structures of the semirings in some subvarieties of(?)I are studied by using the congruences on a semiring.In chapter four,firstly,the subdirect product decompositions of the members in some quasivarietics of the idempotent semirings and the Mal'cev product decompositions of these quasivarieties are given by means of the congruences on an idempotent semiring;secondly, the closed relationships between their subdirect product decompositions and the sturdy frames are characterized.In chapter five,let S be a semiring with a completely regular multiplicativc scmigroup.The equivalent characterizations of which the Green relations L and D are congruences on S are given.Moreover,the result that D is a congruence on S if and only if L and R are congruences on S is proved under certain conditions by using the method of idempotent elements.In chapter six,the concepts of rectangular ring and rectangular Clifford semiring are introduced extending the concepts of left ring and left Clifford semiring and their properties and structures are studied;further,the necessary and sufficient condition that rectangular Clifford semiring is a strong distributive lattice of rectangular ring is got.