Research On Additional Operator Withresiduated Lattices And Its Filter Theory

In recent years, All kinds of t-norms-based fuzzy logic formal systems which are establishmented, the corresponding algebraic structures are special residuated lattices. With in-depth stdudy of fuzzy logic systems, fuzzy logic systems with adding negation operator and ? operator have proposed?such as SBL?, MTL?, MTL??. These logic systems with additional operators are naturely corresponding with residuated lattices with additional operators. In addition, the theory of rough sets also involves additional operator with residuated lattices ?such as Topological operator, Interior operator, closure operator?. This article from the basic structure of residuated lattices, research on all kinds additional operator with residuated lattices. mainly including Extended?Topological? residuated lattices, MV-type triangular algebra, HW-algebra and with vt operator of Non-associative residuated lattices and so on.The main results of this thesis are:(1) The concept of N-filter of extended residuated lattice is introduced and the quotient algebra is constructed, The necessary and sufficient condition for extended residuated lattice become regular residuated lattice is given; the concept of well topological residuated lattice is introduced, its vfilter and basic properties are discussed; on the background of Rough set which is based on extended residuated lattice, the concept of topological extended residuated lattice and its TN-filter are introduced, and the equivalent of the Nv-filter of well topological residuated lattice and its TN-filter is proved.(2) Strong pseudo-involution residuated lattices is given, and adding involution to non-commutative residuated lattices is Strong pseudo-involution residuated lattices is pointed out; MV-type triangular algebra is introduced, and a special kind of MV-type triangular algebra which is Boole algebra based on Interval-valued residuated lattice is constructed; At the same time, the concept of MV-filter of Triangular algebra is given, and the necessary and sufficient condition for triangular algebras become MV-type triangular algebra when each filter of triangular algebras is MV-filter is proved.(3) The notion of filter of HW-algebra is given and the quotient algebra is studied, Boole filter and G-filter of HW-algebra are introduced and the relationship between them are studied; The filter theory and the quotient algebra of RL?- algebra are constructed. (4) The filter theory of Non-associative residuated lattices is studied, the notion of its Boole filter is introduced and its basic properties are studied; At the same time, the vt operator is introduced in the Non-associative residuated lattices, its v filter theory is studied, and there exist a one-to-one correspondence between its v filter and its congruence is proved.