Studies On Some Topics Of Ordered Semihypergroups

Hyperalgebraic structures represent a natural generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in a hyperalgebraic structure, the result of this composition is a subset of a universe set. At present, the theory of ordered semihypergroups is one of the most active ?elds in the research of hyperalgebraic structure. This thesis lies in combining the theories developed from both hyperstructure studies and classic ordered semigroups together, and studying some problems in the theory of ordered semihypergroups. This thesis contains six chapters.In Chapter 1, the research background and recent development on ordered semihypergroups are introduced. Then we analyze the motivations of this thesis, and introduce some basic de?nitions, terminologies and notations that we shall use in this thesis. Finally, we present the main results of this thesis.In Chapter 2, we mainly study the hyperideals of ordered semihypergroups. In particular, we introduce the concepts of completely prime, prime, weakly prime, completely semiprime and semiprime hyperideals in ordered semihypergroups, and establish the relationships among them. Furthermore, we consider the extensions of hyperideals in commutative ordered semihypergroups, and then give some characterizations of hyperideal extensions in terms of n-prime and n-semiprime hyperideals.In Chapter 3, we ?rst introduce the concepts of hypersemilattice strong congruences and complete hypersemilattice strong congruences on ordered semihypergroups,and investigate their related properties. Furthermore, we introduce the notions of left hyper?lters, right hyper?lters and hyper?lters of an ordered semihypergroup, obtain the relation of hyper?lters and completely prime hyperideals, and then study the properties of strong congruences induced by the generated hyper?lters in an ordered semihypergroup. Finally, we introduce the hyper version of Green’s relations in ordered semihypergroups by the generated hyperideals, and discuss their related properties. Especially, we characterize the structure of a-maximal hyperideals of an ordered semihypergroup by terms of the Green’s relation J.In Chapter 4, we ?rst introduce the concepts of order-congruences and strong order-congruences on an ordered semihypergroup, obtain the relationships between strong order-congruences and pseudoorders, and give out homomorphism theorems of ordered semihypergroups by pseudoorders. Furthermore, the notion of ρ-chains in ordered semihypergroups is introduced, and a characterization theorem of the(strong)order-congruences by this notion is given. Then we provide a method of constructing order-congruences. By this constructing method of order-congruences, we answer to the open problem given by B. Davvaz et al. Finally, we de?ne and study the strong order-congruence generated by a strong congruence on an ordered semihypergroup.In Chapter 5, we study the fuzzy hyperideals of ordered semihypergroups in detail. The concept of ordered fuzzy points of ordered semihypergroups is introduced.By using the pointwise ideas and methods, we study the properties of fuzzy left, right and two-sided hyperideals in an ordered semihypergroup, and give some characterizations of semisimple ordered semihypergroups in terms of fuzzy hyperideals. Furthermore, we de?ne and discuss the fuzzy hyper?lters of ordered semihypergroups,and give a wonderful characterization theorem of fuzzy hyper?lters by the completely prime fuzzy hyperideals.In Chapter 6, we extend the concept of quasi-hyperideals of semihypergroups to ordered semihypergroup theory, introduce the notion of quasi-hyperideals of ordered semihypergroups, and investigate its related properties. Furthermore, we de?ne and study fuzzy quasi-hyperideals of ordered semihypergroups. In particular, we give the characterization of fuzzy quasi-hyperideal generated by a fuzzy subset in an ordered semihypergroup. In the sequel, characterizations of some special types of ordered semihypergroups by the properties of fuzzy quasi-hyperideals are given.