| In this thesis,we consider some relative substructures of ordered hypersemigroups.The concepts of relative one-side hyperideals and relative hyperideals of ordered hypersemigroups are proposed.Furtherly,the relatively prime hyperideals,relatively weakly prime hyperideals and relatively semi-prime hyperideals of ordered hypersemigroups are defined.As generalizations of one-side ideals of ordered hypersemigroups,relative bi-hyperideals,relative quasihyperideals,relative(m,n)-hyperideals and relative(m,n)-quasi-hyperideals are given accordingly.In Chapter 1,it mainly introduces the research background of relative structures of semigroups,ordered semigroups and ordered hypersemigroups.Besides,We sketch out the concept of hyperideals of ordered hypersemigroups and relative ideals of ordered semigroups,and give the basic preparation knowledge needed in Chapter 2,Chapter 3 and Chapter 4.In Chapter 2,the concepts of relative one-side(left and right)hyperideals and relative hyperideals of ordered hypersemigroups are introduced,and the related properties and theorems are obtained.Weakening relative condition,we define weakly relative hyperideals,and the Green’s relation based on the weakly relative hyperideals of ordered hypersemigroups is given as an extension of the Green’s relation based on ideals of the semigroups.Furthermore,the concepts of relatively prime hyperideals,relative regular and relative intra-regular are introduced.The equivalent conditions of relatively weak prime are obtained,and the ordered hypersemigroups are described by using relative weakly prime hyperideals and relative intra-regular.Finally,the concepts of relative hyperideals of Cartesian product of two ordered hypersemigroups and direct product of many ordered hypersemigroups are studied and their related conclusions are given.In Chapter 3,the concepts of relative bi-hyperideals and relative quasi-hyperideals of ordered hypersemigroups are put forward.And the concepts of relatively prime bi-hyperideals of ordered hypersemigroups are explored by analogy with the properties of ideals of semigroups.The connection between relative bi-hyperideals and relative quasi-hyperideals is obtained.The equivalent characterizations of relatively semi-prime bi-hyperideals and relatively regular ordered hypersemigroups are given.As a special kind of relative structure of ordered hypersemigroups,the concepts and some properties of relative A-hyperideals and relative A-quasihyperideals are studied in this chapter.In Chapter 4,we come up with relative(m,n)-hyperideals of ordered hypersemigroups,and explore the generation and properties of relative(m,n)-hyperideals;The associated theorem of(m,n)-regular ordered hypersemigroups,(m,0)-hyperideals,(0,n)-hyperideals are then given;(m,n)-quasi-hyperideals are generalization of one-side hiperideals.We introduce the concept of(m,n)-quasi-hyperideals and its generated form;Furtherly,after defining the left Hm-hyperideals and right Hn-hyperideals,we study connection about the two and relative(m,n)-quasihyperideals,namely intersection property of relative(m,n)-quasi-hyperideals;The concept of weakly relative(m,n)-simple is proposed and used to describe ordered hypersemigroups. |
