Research On The WMTL-filter And Related Properties On The Residual Lattice

Residuated lattices are recognized as one of the most widely logical algebraic structures in fuzzy logic theory.This paper gives a further study of residuated lat-tice.The theory of filter on the residuated lattice is studied further first.The filter plays an important role in studying logical algebras and the completeness of the cor-responding non-classical logics,the research work in this field is very popular among scholars of all ages.With the introduction of the concept of WMTL-filter on the residuated lattice,the corresponding properties is researched and the relationship between the WMTL-filter and other filters is studied;Secondly,in order to know whether the set of all regular elements in the residuated meet semi-lattice is meet semi-lattice or not,the definition of residuated meet semi-lattice is given,and prop-erties of residuated meet semi-lattice is discussed;In addition,some relationships between residuated lattice and residuated meet semi-lattice is given.The concrete contents of this paper are as follows:Chapter 1:Preliminaries.We give the concepts and correlation conclusions of poset,lattice,residuated lattice,the filter of residuated lattice,MTL algebras,BL algebras,WMTL-algebras.Chapter 2:WMTL-filter in residuated lattice and its relevant properties.First-ly,the concept of WMTL-filter is introduced in residuated lattice,some character-izations are obtained.It turns out that the WMTL-filter is a special kind of filter in residuated lattice;Secondly,the equivalent characterizations of WMTL-filter are given,and it is proved that the quotient algebra of a residuated lattice induced by a WMTL-filter is a WMTL-algebra;Finally,the relations between WMTL-filter and the implication filter,positive implicative filter,MTL-filter,2-MTL-filter are discussed systematically.Chapter 3:The residuated meet semi-lattice and its relative properties.Firstly,the definition of residuated meet semi-lattice is given.Through the further study of its properties,it is proved that the set of all regular elements in the residuated meet semi-lattice is meet semi-lattice,and an example is given to illustrate the set of all regular elements in the residuated meet semi-lattice is not its substructure.Secondly,it is proved that the regular residuated meet semi-lattice which satisfies residuated commutative law:x(?)(x?y)= y(?)(y?x)is Wajsberg algebra.Finally,by the residuated commutative law:x(?)(x?y)= y(?)(yy?x),we obtain that L is a BL algebra,if and only if L is MTL algebra satisfying residuated commutative law.