In the sense of cutset, the relation between the rough set of a cutset and the cutset of a rough set, a fuzzy rough set of a fuzzy primary ideal in a semigroup is proved to be its fuzzy primary ideal. The sixed chapter is Rough Primary Ideals and Rough Fuzzy Primary Ideals in rings, the corresponding results of a semigroup are generalized in a ring. The last chapter is Rough-fuzzy Subsemirings in Semirings, First, the lower and upper rough-fuzzy semirings together with the left and the right ideal are defined.

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According to the existence results of general equilibrium problems and vector equilibrium problems have been studied more and more. Inspired and motivated by these research results, this paper is devoted to study systematically a class of equilibrium problems, which is unify and extension of a large number of known equilibrium problems and variational inequalities problems. The research is carried on from three aspects.Firstly, in finitely continuous topological spaces, we introduce four new types of the system of generalized vector quasi-equilibrium problems, and we derive some existence results of a solution for the system of generalized vector quasi-equilibrium problems via the maximal element theorems in product finitely continuous topological spaces.Secondly, in complete metric spaces, we provide the Ekeland variational principle to equilibrium problems with set-valued maps. And via the Ekeland variational principle, existence results for vector equilibrium problem with set-valued maps and the system of vector equilibrium problem with set-valued maps.

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From the point of view of measure theory, this paper gives the unified definition of measurable real fuzzy set-valued function and n-dimensional measurable real fuzzy set-valued function, and discusses the fundamental properties of them and their invers fuzzy set-valued function by combining with fuzzy σ-algebra.

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And the sufficient and necessary conditions of the monotone set function which is continuous of the empty set and the full set are given.

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University ,Changsha 410075,HunanAbstract: In view of the deficiency in methods for choosing zonation indicators of mudflow danger degrees,a new method for choosing zonation indicators of mudflow danger degrees is put forward based on the rough set theory in this paper.Firstly,according to the environmental conditions of forming mudflows,an indicator choice system is established.Then the indicators for choice are discretized by indicator grading.Finally,based on the discernable matrix of the rough set,a core set of the indicators is found and the representative indicators are sieved out.

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In this dissertation, we try to srudy the organic combination of the t-norm operator set, rough set and fuzzy set.

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Based on the method of multi-objective fuzzy optimization and the overhead traveling crane design, a mathematical model of fuzzy optimization for main box beam section is established. A fuzzy goal anthology is constructed. The fuzziness of multi-objective function and constraints is defined. The transformation of the fuzzy constraints set to normal constraints set in real number domain is realized by applying optimal level cut set method. The optimal solution and optimal constraint value of sub-objective function in the feasible field are found using the genetic algorithms.

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