Study On The Relation Between Lattice Implication Algebras And Correlative Logic Algebras

Logic algebra is the logical foundation of reasoning mechanism in many fields such as computer science,information science,cybernetics and artificial intelligent. Lattice implication algebra is a kind of logic algebra,and it is the basis for researching on lattice-valued logic theory.While the purpose of researching on lattice-valued logic is to provide a kind of logical foundation for uncertainty reasoning and automated reasoning.Lattice implication algebra has quite close relations between many other logic algebras such as MV-algebras,BCK-algebras and R₀-algebras.The author mainly researched into the relations between lattice implication algebras and some logic algebraic systems including regular Fuzzy implication algebras,MTL-algebras,IMTL-algebras,WNM-algebras,NM-algebras, BL-algebras,implication lattices,Heyting-algebras and De Morgan-algebras.At the same time,the author also studied the dual properties of lattice implication algebras.At last,the author introduced the concept of strong prime LI-ideals of lattice implication algebras and received its some properties.All the above are the problems that are paid much attention to in lattice-valued logic and uncertainty reasoning theory.This paper mainly obtained the following results:1.Studied the relation between lattice implication algebras and regular Fuzzy implication algebras.By defining two operations∨and∧in a regular Fuzzy implication algebras,such that it can be a lattice structure,then proved that such regular Fuzzy implication algebra is a lattice implication algebra if it satisfied certain condition,and the condition is sufficient and necessary.2.Gained the relations between lattice implication algebras and MTL-algebras, IMTL-algebras,WNM-algebras,NM-algebras,BL-algebras,implication lattice, Heyting-algebras,De Morgan-algebras,respectively.3.By talking about the properties of filters and LI-ideals of lattice implication algebra,depicted the dual property of lattice implication algebra.4.Introduced the notion of strong prime LI-ideals(briefly,SPLI-ideals)of lattice implication algebras,investigated the relations between SPLI-ideals and prime LI-ideals,between SPLI-ideals and maximal proper LI-ideals,between SPLI-ideals and the finite union property,and between ultra-filter and SPLI-ideal, concluded that SPLI-ideals are equivalent to maximal proper LI-ideals.Extended the results already received.