On Lattice Implication Algebras And Structure Of Generated Filter

Lattice implication algebra as one of the theoretical basis of lattice-valued logic and uncertainty information processing, which is a very important logical algebra, and it’s filter just reflects the MP rule in logical reasoning. The research of it’s nature, structure, filter’s structure and the relationship of filters were got a lot of research by many scholars at home and abroad, and fruitful.Based on the former outcomes, the current dissertation discusses some properties of lattice implication algebra and it’s filters, mainly studied the structural characteristics of the generated filter. This helps to recognize the structural and nature of the filters, and sort out the inference rules to a certain extent. The major achievements include the following three aspects:1. Further discussed some properties of lattice implication algebra itself and related properties of it’s filters. By defining the structure of Fa, got the necessary and sufficient condition of filters become implication filters in a lattice implication algebra, obtained some equivalent conditions about the positive implication filters.2. Studied the properties of generated sub-lattice and generated lattice filter, obtained the generated structure of lattice filter by the non-empty subset and corresponding relations of the union and join of non-empty subsets of a lattice. Obtained the generated structure of the union of two filters of lattice implication algebra. Got the generated structure of the infimun of two elements of a lattice H implication algebra. Gave the concepts of interval-valued fuzzy filter and interval-valued fuzzy generated filter in a lattice implication algebra and some properties of them. Obtained all the interval-valued fuzzy generated filters (F F(L),∨,∧) in a lattice implication algebra is a complete distributive lattice.3. Gave out some properties of generated sub-lattice and generated lattice filter on limited chain Lm, using the properties of atom and dual atom on limited lattice implication product algebra, studied the form of filter generated by atom and dual atom on limited lattice implication product algebra, and got the structure of filter generated by dual atom and atom on three kinds of limited lattice implication product algebras. Also analysis of the differences of filters generated by atom and dual atom.