Research On Some Problems In Super-order Structure

In this thesis,we mainly study some problems of hyperorder structure.On the one hand,we study notions of ideals,derivations,filters,prime ideals,prime filters,fuzzy ideals,fuzzy filters,fuzzy prime ideals,fuzzy prime filters in meet-hyperlattices and Fuzzy prime ideal theorem and its related properties on meet-hyperlattices.On the other hand,we give two equivalent characterizations of the regular,inter-regular and semisimple(fuzzy)ordered hypersemigroups in terms of ideals.In addition,by deeply studying the ideals,congruences and representation theorems on lattice-ordered semigroups,we generalize them to(meet)hyperlattice-ordered semigroup,and give the representation theorem of a special hyperlattice-ordered semigroup.Finally,further use of S-action,we generalize the relevant theories on S-lattice to S-hyperlattice(meet-hyperlattice),giving the concept of S-hyperlattice and introducing properties of congruences of S-hyperlattice.The specific layout is as follows:In the first chapter,we mainly introduce the research background,research status and significance of hyperorder structure.At the same time,the main work of this thesis is given.In the second chapter,we introduce ideals and congruences induced by derivations and properties on(dual)distributive meet-hyperlattices.In the third chapter,we give the equivalent characterization of the regular,inter-regular and semisimple(fuzzy)ordered hypersemigroups in terms of ideals.In the fourth chapter,we introduce definitions of fuzzy ideals,fuzzy filters,fuzzy prime ideals,fuzzy prime filters,fuzzy congruences,and proof the related theorem on meet-hyperlattices.In the fifth chapter,we give the representation theorem of a special kind of hyperlatticeordered semigroups and the notion of S-hyperlattice and the concept of S-hyperlattice congruence based on S-lattice and S-lattice congruence.Also we study the relationship between S-hyperlattice congruence and S-hyperlattice pseudo-congruence on S-hyperlattice,and related properties.