Research On Some Problems Of Superplastic Semigroups

The structures of ordered hypersemigroups are investigated in this paper. Hypersemigroups are studied by their(m, n)-hyperquasi-ideals, m-left hyperideals and n-right hyperideals when the orders of ordered hypersemigroups are trivial ordered. Then the different properties and the relationships among(m, n)-hyperquasi-ideals, m-left hyperideals and n-right hyperideals are presented. Ordered hypersemigroups are investigated via hypercongruences. The definition and properties of regular hypercongruences of ordered hypersemigroups are introduced. Then, the concept and equivalent conditions of strongly regular hypercongruences are given on the basis of regular hypercongruences. In the sequel, the special regular hypercongruences–strongly order hypercongruences are given.In first part, the research background and some basic knowledge on this thesis are introduced.In second part, firstly, the concepts of(m, n)-hyperquasi-ideals, m-left hyperideals, n-right hyperideals and the relationships among them are introduced. The form which are generated by a nonempty subset of a hypersemigroup and the intersection properties of(m, n)-hyperquasiideals, m-left hyperideals and n-right hyperideals are given. Secondly, the concepts, intersection properties and equivalent conditions of minimal(m, n)-hyperquasi-ideals, minimal m-left hyperideals and minimal n-right hyperideals are presented. Finally, the notions and some characterizations of(m, n)-quasi-simple, m-left simple and n-right simple hypersemigroups are given.In third part, firstly, the concept of regular hypercongruences is introduced. Then, a hypercongruence σ of ordered hypersemigroups is given by hyperideal such that quotient hypersemigroup S /σ is an ordered hypersemigroup. And it can solve the existed problem of regular hypercongruences by B. Davvaz, etc. In further, the relationships between regular hypercongruences and pseudoorders and the minimal regular order are discussed. Secondly, the definitions of strongly regular hypercongruences and quasi-chain modulo σ are introduced on the basis of regular hypercongruences. Then the properties of quasi-chain modulo σ are investigated and the characterizations of regular hypercongruences are presented by quasi-chain moduloσ. In further, the necessary and sufficient conditions of regular hypercongruences and strongly regular hypercongruences are given. Moreover, the properties of l-hypercongruences and l-hypersemigroups are introduced and a result that the l-hypersemigroups of l-hypercongruences are regular hypercongruences is proved. Finally, the definition of strongly order hypercongruences is presented. The properties and equivalent conditions of strongly order hypercongruences are discussed, and an important conclusion that the strongly order hypercongruences of ordered hypersemigroups are regular hypercongruences is given.