Studies On Some Topics Of Semigroups And Semihypergroups

As an important branch of algebra,the classical(ordered)semigroup algebra theory has now become a hot-topic subject.There are mature research methods and perfect theoretical system in(ordered)semigroup theory.(Ordered)semihypergroup represent a reasonable generalization of(ordered)semigroup theory.At present,the theory of(ordered)semihypergroups is one of the most active research topics in the research of hyperalgebraic structure.This thesis aims to apply the theory of hy-peralgebraic structure to(ordered)semigroup algebra,introduce a unary involution operation,and study some related problems in the theories of(ordered)semigroups and(ordered)semihypergroups.This thesis contains seven chapters.In Chapter 1,the research background and recent development on semigroup and(ordered)semihypergroups are introduced.Then we analyze the motivations of this thesis,introduce some basic definitions and notations that we shall use in this thesis.Finally,we present the main results of this thesis.In Chapter 2,we firstly introduce the concept of Q-inversive semigroups and char-acterize group congruences on this class of semigroups in terms of Q-full and weakly Q self-conjugate subsemigroups.Furthermore,a construction of all Q-subdirect prod-ucts of Q-inversive semigroups is obtained.Finally,we study the Q-unitary covers for Q-inversive semigroups.In Chapter 3,we study the ordered*-semigroups in terms of fuzzy subsets in detail.To begin with,we introduce the concepts of prime,weakly prime and semiprime fuzzy ideals of S.Furthermore,we establish the relationships among these three types of fuzzy ideals and give some characterizations of the intra-regular ordered*-semigroups in terms of fuzzy ideals.Finally,we study the fuzzy filters and discuss the relationships between the filters and the fuzzy filters in ordered*-semigroups.In Chapter 4,we firstly introduce the concept of filters of an ordered(*,?)-semigroup,and investigate some related properties.Furthermore,we construct a complete semilattice congruence N on an ordered(*,?)-semigroup M in terms of the principal filters and give some characterizations of intra-regular ordered(*,?)-semigroups.Finally,for any a?M,the relationship between the congruence class(a)_Nand the principal filter N(a)generated by a is established.In Chapter 5,we firstly give a negative answer to the open problem introduced in[K.Hila,B.Davvaz and J.Dine,Study on the Structure of?-Semihypergroups,Communications in Algebra 40(2012),2932–2948].Furthermore,we introduce the concept of weakly absorbent?-semihypergroups,and solve the other problem inves-tigated in the same paper as to the above-mentioned.Finally,we study the Green's relations on(*,?)-semihypergroups and provide some characterizations of ordered semihypergroups which are nil-extensions of simple ordered semihypergroups.In Chapter 6,we firstly extend the concept of pseudoorders of ordered semi-hypergroups to ordered*-semihypergroups.Furthermore,we introduce the concept of weak pseudoorders and construct a regular equivalence relation on an ordered*-semihypergroup such that the corresponding quotient structure also is an ordered*-semihypergroup.Meanwhile,we obtain a homomorphism theorem of ordered*-semihypergroups by weak pseudoorders.As an application of this result,we com-pletely solution to the open problem introduced in[B.Davvaz,P.Corsini and T.Changphas,Relationship between odered semihypergroups and ordered semigroups by using pseuoorders,European J.Combin.,44(2015),208–217].