The Algebraic Structure Of Two Classes Of Semigroups

The algebraic structure of a semigroup is one of the important topics in the study of semigroup theory.This thesis mainly studies the algebraic structure of fuzzy soft inverse semigroups and H*-commutative semigroups.This thesis consists of five chapters.In Chapter 1,we introduce the research background and current situation of semigroups,fuzzy sets,soft sets and fuzzy soft sets,as well as the structure of this thesis.In Chapter 2,we give some basic concepts and relevant conclusions which are frequently used in this thesis.In Chapter 3,by applying soft set theory to fuzzy congruences and fuzzy inverse subsemigroups on inverse semigroups,we introduce the notions of fuzzy soft inverse semigroups and normal fuzzy soft inverse semigroups on inverse semigroups.Then,we investigate some important properties of such semigroups.Subsequently,the one-to-one correspondence between fuzzy soft congruences and normal fuzzy soft inverse semigroups on inverse semigroups is established.Further,we study algebraic properties of several important kinds of fuzzy soft congruences on inverse semigroups.Finally,by using θ-soft cut sets and θ-soft cut relations,the equivalent characterization of fuzzy soft congruences,fuzzy soft inverse semigroups and normal fuzzy soft inverse semigroups on inverse semigroups is obtained.It is proved that a fuzzy soft set(f,A)is a fuzzy soft inverse semigroup on an inverse semigroup S if and only if(f,A)_θ is a soft inverse semigroup on S.In Chapter 4,we generalize the concepts of R-commutative semigroups,L-commutative semigroups and H-commutative semigroups which are introduced by P.G.Trotter.The concepts of R*-commutative semigroups,￡*-commutative semigroups and H*-commutative semigroups are firstly introduced,in the context of Green*relation.Then,several important properties of such semigroups are investigated.In particular,it is proved that a semigroup S is an H*-commutative semigroup if and only if H*is a commutative congruence on S.Finally,we show that every H*-commutative semigroup is decomposable into a semilattice of H*-archimedean semigroups.In Chapter 5,we summarize the whole work and put forward the prospects for further.