Study On The MP~M Medium Algebras

The medium logic is founded on medium principle(three valued:true,middle and false) in the 80's twenty century by professors Zhu wu-jia and Xiao Xi-an.From then on,it has been investigated widely in syntax,sematic and formalication.The calculus system of mediun logic is constituted with propositional calculus system of mediun logic MP and an extension of MP called MP*,predicate calculus system of mediun logic MF and an extension of MF called MF*,predicate calculus system with equality symbol"="of mediun logic ME,ME*.The MP~M system is a 3-valued logic propositional calculus system which is based on propositional calculus system of mediun logic MP and MP*,and it is used to deal with the incomplete information in the database.With several years investigated,it has been got some outcomes,an- d it applied to query optimization. The MP~M system also has lots of algebraic properties,so a new type of algebra is introduced called the MP~M medium algebra to investigate its properties.The MP~M medium algebra is a new type of algebra,and it appends to a unitary operation"μ",which is based on De Morgan algebra.It has lots of favorable algebraic properties and relationship with some logic algebras.The other scholars proved MP~M medium algebra could transform to be Boole algebra and medium algebra.Itself also is a Kleene algebra.The paper continues to investigate it's properties and prove itslf is a Kleene algebra under general situation,and also studing the relationship with Stone algebra, Nelson algebra,Rough algebra and Fuzzy sets.The main innovative points in the paper have:(1)Discuss the properties of MP~M medium algebra farther and make two examples to illuminate.(2)Prove the MP~M medium algebra itself is a Kleene algebra under general situation,and it also can be a Stone algebra when satisfying some condition,but it can't be a Nelson algebra by making a example to explain.(3)Prove Rough Double Stone algebra,Approximation Space algebra and Rough Nelson algebra are De Morgan algebra,Rough Nelson algebra is also be a Kleene algebra.In succession, with these outcomes can prove Rough algebra become the MP~M medium algebra when chosen proper a unitary operation"μ".(4)Apply Fuzzy sets to the MP~M medium algebra,so called Fuzzy MP~M medium algebra,and discussing some properties of it.