The Characterization Of Congruences On Inverse Semirings

In this thesis, the characterization of some kinds of congruences on additiveinverse semirings are studied. The thesis is divided into three chapters.In Chapter 1, we introduce several stages of the development of semiring theoryand its important position in mathematics, then we make clear the main basis andmethods of research about this topic and give the basis for theories in the paper.In Chapter 2, it investigates mainly the characterization of congruences onadditive inverse semirings. Firstly, we provide the definition of congruence pairs onadditive inverse semirings and give some results about congruences and congruencepairs. Then consider the version of kernel and trace, we have the main conclusionthat congruences on semirings are ascertained by congruence pairs on it uniquely.In Chapter 3, It is dedicated to the study of another characterization of congru-ences on additive inverse semirings. Firstly, we provide a set of independent axiomscharacterizing the kernel of a congruence on an additive inverse semiring and showhow to reconstruct the congruence from its kernel. Next we show that the mappingρ→trρis a complete homomorphism of the congruence lattice of S to the latticeof normal congruences on E+(S), and that the congruence induced by it has all itsclasses complete modular lattices.In Chapter 4, it's devoted to introduce how to characterize the partitions of theidempotents of an additive inverse semiring S which are induced by a congruence onS and provide a characterization of the minimum congruences on S inducing a givensuch partition. The other study obtains the characterization of maximum additiveidempotent-separating congruence on inverse semiring.