Fuzzy Algebra Based On Lattice Valued Logic

The important direction of the research of fuzzy logics is studing the logical algebra system. Based on posets and some operators,logical algebra systems establish the bridge of logic and algebra and use algebraic method to solve the problem. Residuated lattice is an important class of algebraic systems.In this paper, the properties of t-filtes,I-filters,EIMTL-filters in residuated lattice are studied. N-dimensional fuzzy filters, n-dimensional fuzzy EIMTL-filters, n-dimensional fuzzy t-filters, n-dimensional fuzzy congruence relations are researched.The main results obtained in this dissertation are summarized as follows:In Chapter two,we discuss the relationship between t-filters and I-filters.We generalize the notions of EIMTL-filters and associative filters to residuated lattices and study some of their properties.The conditions under which an EIMTL-filter is a t-filter are considered.It is proved that every residuated lattice has a proper EIMTL-filter.In addition, the conditions under which a residuated lattice has a proper associative filter are considered. We also generalize the notions of EIMTL-filters to pseudo-residuated lattices and study some of their properties.In Chapter three, the concept of n-dimensional fuzzy filters on a residuated lattice is introduced and their characterizations are derived.The relationship between λ-upper set of a n-dimensional fuzzy filter and the n-dimensional fuzzy filter is discussed.We also introduce the notion of n-dimensional fuzzy EIMTL-filters on a residuated lattice and derive their characterizations.The relationship between λ-upper set of a n-dimensional fuzzy EIMTL-filter and the n-dimensional fuzzy EIMTL-filter is also discussed.In addition,the concept of n-dimensional fuzzy t-filters on a residuated lattice is introduced and their properties are studied.The definition and characterization of n-dimensional fuzzy divisible filter and n-dimensional strong filter are obtained by using n-dimensional fuzzy t-filter.In Chapter four,the concept of n-dimensional fuzzy congruence relation on a residuated lattice is introduced and its properties is studied.We discuss the relationship between λ-upper cut sets of a n-dimensional fuzzy congruence relation and the n-dimensional fuzzy congruence relation.Then the n-dimensional fuzzy congruence relation corresponding to a given n-dimensional fuzzy filter on residuated lattices is defined.In addition,we discuss the relationship between α-cut relations of a n-dimensional fuzzy equivalence relations and the n-dimensional fuzzy equivalence relation.