On Studies Of Some Ordered Hypersemigroups

This paper mainly considers some problems in some ordered hypersemigroups.At the beginning,we put forward the concepts of sub-strong hyperfilters and saturated sets,then define the concept of weakly saturated ordered hypersemigroups.Next,we introduce the concepts of several classes of commutative ordered hypersemigroups,and study the related properties.Based on weakly saturated conditions,we study the semilattice decomposition of these ordered hypersemigroups.Finally,the concepts and related properties of α-hyperideal are introduced.The following sections detail the content:In the first chapter,we present the development of semigroups,ordered semigroups and hyperstructures,and describe the origin and development of several classes of weakly commutative semigroups and α-ideals.At last,we give the relevant basic knowledge of this paper.In the second chapter,we give the concepts of sub-strong hyperfilter and saturation firstly,and use saturation set to define weakly saturated ordered hypersemigroups.Then,we extend weakly commutative(poe-)ordered semigroups to ordered hypersemigroups,define weakly commutative(poe-)ordered hypersemigroups,and give their related properties.Based on the weak saturation condition,we give the characterization of the sub-strong hyperfilter N(x)of these ordered hypersemigroups,and prove the weakly commutative ordered hypersemigroup is a semilattice decomposition of Archimedean ordered hypersemigroup.In the third chapter,we extend the weakly external commutative ordered semigroups,pseudo commutative ordered semigroups and cyclic commutative ordered semigroups to ordered hypersemigroups,and define the concepts of weakly external commutative ordered hypersemigroups,pseudo commutative ordered hypersemigroups and several classes of cyclic commutative ordered hypersemigroups.Then,we give the characterization of the sub-strong hyperfilter N(x)of pseudo commutative ordered hypersemigroups and weakly cyclic commutative ordered hypersemigroups.Main conclusions of this chapter is that based on the condition of weak saturation,weakly external commutative ordered hypersemigroups are semilattice decompositions of a-connected ordered hypersemigroups,as well as pseudo commutative ordered hypersemigroups is a semilattice decompositions of Archimedean ordered hypersemigroups.In the fourth chapter,we mainly give the concepts and properties of generalized αhyperideal,α-hyperideal and complete α-hyperideal on hypersemigroups.Then we study the equivalent characterization of generalized α-hyperideal is α-hyperideal.