Studies On Some Topics Of Partially Ordered Hypersemigroups

In this article,we prove that the theory of fuzzy partially ordered hypersemigroups is similar to the theory of fuzzy partially ordered hypersemigroups.We gives the equivalent characterizations of the classes of fuzzy ideals on partially ordered hypersemigroups,and give some equivalent characterizations of the classes of regular,quasi-regular and left(resp.right)regular partially ordered hypersemigroups.The equivalent characterization theorems for semisimple partially ordered hypersemigroups are also given in this paper.Finally,we introduce the concept of hyperlattice-ordered semigroups,and introduce the concept of S-hyperlattices imitating the action of ordered groups on the partially ordered set.On the basis of hyperlattice theory,we give the S-hyperactions and representation theorems of hyperlattice-ordered semigroups.In first chapter,we give a brief introduction to the background of this article and the basic knowledge needed to be used later.In second chapter,the first section we introduce the classes of fuzzy ideals on partially ordered hypersemigroups.We gives several equivalence definitions of the fuzzy left(right)ideals,fuzzy quasi-ideals,fuzzy bi-ideals of partially ordered hypersemigroups,based on fuzzy sets.The second section we gives the equivalent characterization theorems of the classes of fuzzy ideals mentioned in the first section in terms of fuzzy subsets of ordered hypersemigroup.In third chapter,the first section we give the equivalence definitions for the classes of regular partially ordered hypersemigroups.We obtain the equivalent characterizations of the regular partially ordered hypersemigroups,the intra-regular partially ordered hypersemigroups and the left(right)quasi-regular partially ordered hypersemigroups,in terms of fuzzy sets.The second section are give other equivalent characterizations for the classes of regular partially ordered hypersemigroups mentioned in the first section.In fourth chapter,we introduce the definition of semisimple partially ordered hypersemigroups and give some equivalent propositions.Finally,we give the equivalent characterizations of semisimple partially ordered hypersemigroups in terms of fuzzy subsetof them.In fifth chapter,we study the hyperlattice-ordered semigroups,the first section we introduce the concepts of hyperlattice-ordered semigroups.We give the concepts of hypercongruences and their related properties on hyperlattice-ordered semigroups.Based on these,we obtain some properties of sl-hyperideals on hyperlatticeordered semigroups.Then,we give out the homomorphism basic theorems on the distributive hyperlattice-ordered semigroups.The second section we introduce the actions of hyperlattice-ordered semigroups on hyperlattices,this action is called as S-hyperlattices.We also research the related properties and the representation theorem of hyperlattice-ordered semigroups,that is,any hyperlattice-ordered semigroup is homomorphism with a hyperlattice-ordered semigroup that is constructed of auto-homomorphisms on the corresponding hyperlattice.Then,we introduce the concepts of the S-hyperlattice hypercongruences on S-hyperlattices and give out generated theorems.Finally,we introduce the S-hyperlattice hyperideals on the S-hyperlattices and prove that a down-convex subset of S-hyperlattice is a S-hyperlattice hyperideal.