| This dissertation comes from "the natural science research projects of Education Department of HenanProvince"(No.2009A520015)"The Algebraic Structure of Interval-valued Fuzzy Logic" and "the keyscientific and technological project of Henan Province"(No.092102210149)"Research on Flexible ControlModel and its System based on Interval Structure".Fuzzy set was proposed by American computer and automatic control theory expert Zadeh in1965.Fuzzy sets theory had been widely applied to a range of specific areas. In1986, Atanassov introduced thenotion of intuitionistic fuzzy sets, which is an expansion of the Zadeh's fuzzy sets. By adding a newattribute parameter non-membership function in intuitionistic fuzzy sets, the neutral state of "non-this andnon-that" can be described and the fuzzy characteristics of the objective world is more delicately portrayed.It can more delicately depict the fuzziness of objective things, and better reflect the ambiguity of objectiveworld. The research of intuitionistic fuzzy sets has caused many scholars' attention. The research ofintuitionistic fuzzy reasoning is an important part of intuitionistic fuzzy sets, and intuitionistic fuzzyimplication is the key to the research of intuitionistic fuzzy reasoning. However, the constructive researchof the intuitionistic fuzzy implication is less. According to the implication and basic algebra in thepoint-valued fuzzy logic, and the ordering relation on the intuitionistic fuzzy sets, the in-depth study ofimplication is extended to the intuitionistic fuzzy sets.BL-algebra was introduced by Hajek in1996, it is an algebraic structure of Basic Logic, and plays animportant role in studying logic. MV-algebra, Godel-algebra and product algebra are particular cases ofBL-algebras. Filters theory plays an important role in studying these logical algebras. Since from logical point of view, a kind of filter corresponds to a set of provable formula. And it provides a general method tostudy algebraic system structure, i.e., filters theory is directly applied to quotient algebra structure. Withdevelopment of intuitionistic fuzzy sets, people research the filters in combination with algebras andintuitionistic fuzzy sets.Based on the above theory, the principal work accomplished is as follows:Firstly, a new intuitionistic fuzzy implication is reconstructed based on the intuitionistic fuzzy sets,and its regularities, monotonicities and important algebraic properties are discussed.Secondly, intuitionistic fuzzy sets applie to the conception of BL-algebras filter. Intuitionistic fuzzyfilter and intuitionistic fuzzy lattices filter are introduced, respectively. Intuitionistic fuzzy filter's importantproperties are investigated. We research the relation between the intuitionistic fuzzy filter and intuitionisticfuzzy lattices filter, i.e., any intuitionistic fuzzy filter is an intuitionistic fuzzy lattices filter. Further, wegive the relation between the intuitionistic fuzzy filter and fuzzy filter.Thirdly, in the basis of intuitionistic fuzzy sets, we give the notion of intuitionistic fuzzy prime filterof BL-algebras and investigate its properties. We provide the intuitionistic fuzzy prime filter extensiontheorem.Fourthly, in the basis of intuitionistic fuzzy sets, Boolean filter, implicative filter, positive implicativefilter, ultra filter and obstinate filter are introduced in the intuitionistic fuzzy sets, respectively. And theirimportant properties are investigated. The intuitionistic fuzzy Boolean filter is proved to be equivalent tothe intuitionistic fuzzy implicative filter, the intuitionistic fuzzy ultra filter is also proved to be equivalent tothe intuitionistic fuzzy obstinate filter, and that every intuitionistic fuzzy Boolean filter is an intuitionisticfuzzy positive implicative filter, but the converse may not be true in the BL-algebras. Furthermore, theconditions under which an intuitionistic fuzzy positive implicative filter is an intuitionistic fuzzy Boolean filter are constructed. |
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