| The fuzzy logic has widespread application inindustrial control, artificial intelligenceresearches, uncertain reasoning and decision support. Residuated lattice is the algebraicstructure of fuzzy logic formal systems, and has achieved abundant research results. With thefuzzy logic theory intensive studied, the non-associative fuzzy logic structure attracts greatattention by all countries’ the scholars, and becomes a hot theory. Non-associative residuatedlattice is an algebraic abstract of the non-associative fuzzy logic formal systems. The researchof which has made some progress in recent years. On this basic, this thesis will research somebasic properties of non-associative lattice and some related filters theory.The main results of the thesis are:(1) Give the complete classification of low order non-associative lattice withMathematics software. The notions of prelinear non-associative residuated lattice andinvolution non-associative residuated lattice are introduced, and their basic properties arestudied. It is proved that every prelinear non-associative residuated lattice is a boundeddistributive lattice, and a necessary and sufficient condition for non-associative residuatedlattice to be prelinear is given. At the same time, the basic properties of involution and stronginvolution non-associative residuated lattice are studied, and the equivalence conditions forinvolution non-associative residuated lattices and strong involution non-associative residuatedlattices are given respectively. Finally a counter example is given to show that there is astrong involution and prelinear non-associative residuated lattice which is not an implicationlattice.(2) It is proved that the two definitions of the filters are equivalence. The notion ofregular filter and MV-filter are given in the non-associative residuated lattice, and thenproperties and sufficient and necessary conditions of them are studied. The notion of fuzzyfilter is given, and the characteristic of fuzzy filter is described by level subset and filters. Atlast, on the basic of fuzzy filter theory, the corresponding quotient algebra is structured, andthe basic theorem fuzzy homomorphism is proved.(3) Apply soft set to (non-associative) residuated lattice and its filter theory. Softresiduated lattice and filteristic soft residuated lattice are introduced, and some operation,propertie and the corresponding examples are studied. The notion of fuzzy (regular) filter isgiven, and whose necessary and sufficient condition is discussed. Finally the properties and operations of fuzzy (regular) fileristic soft residuated lattice are considered. |
