| As an important tool for studying non-classical logic systems,the researc h on various kinds of logic algebras has been paid close attention by scholars.EBL-algebras are unbounded generalization of BL-algebras,which can be connected with BL-algebras through idempotent elements.Filters play a vital role in the study of logical algebras.In this paper,the filters and fuzzy filters in EBL-algebras are deeply studied.The main contents are as follows.Firstly,according to the characteristics of EBL-algebras,the equivalence definition of filters expressed by implicative symbols is proved by idempotent elements.On this basis,the prime filters and Boolean filters in EBL-algebras are defined,the equivalent characterization of prime filters and the equivalence theorem of Boolean filters constructed by quotient algebras are given,and the conclusion that maximal filters and prime filters are equivalent under the premise of Boolean filters is obtained.Secondly,the concepts of implicative filters,positive implicative filters and fantastic filters are introduced respectively,and it is proved that they are all filters.A series of equivalent characterizations are also given,and some relational theorems such as Boolean filters are implicative filters and positive implicative filters are fantastic filters are obtianed.At the same time,the equivalence between Boolean filters and positive implicative filters,and the equivalence between Boolean filters and implicative and fantastic filters are proved.Then,with the help of the level subset of fuzzy sets,the concept of fuzzy filters is defined in EBL-algebras,and the related equivalent characterizations and inferences are discussed.At last,the concepts of fuzzy prime filters,fuzzy Boolean filters,fuzzy implicative filters,fuzzy positive implicative filters and fuzzy fantastic filters are introduced,their properties and equivalence theorems are studied,and the conc lusion that fuzzy Boolean filters are fuzzy implicative filters is proved.The conditions under which fuzzy implicative filters become fuzzy Boolean filters are given,and the equivalent relationships among fuzzy Boolean filters,fuzzy positive implicative filters and fuzzy implicative and fantastic filters are obtained. |
