Partial Order On The Set Of Implication Algebra,

Non-classical logic is the theoretical basis of many-valued logic,fuzzy reasoning and fuzzy control. Fuzzy logic is the most active branch of non-classical logic. Fuzzy logic is studied with algebraic tools in this paper. A kind of algebraic abstract of fuzzy logic,Implication Algebra on a partial ordered set,is given. The relations between Implication Algebra and other algebraic structures,such as MV-Algebra and Heyting Algebra etc.,and the filter and the structure of Implication Algebra on a partial ordered set are studied. Difference from other algebraic structures which are introduced for some logic system,Implication Algebra is a abstraction of one logic connective,i.e. implicative operator,and other operators in it are all introduced by implicative operator. The main results of this paper is given as the following:1. The concept of Basic Implication Algebra and Implication Algebra on a partial ordered set are obtained by studying the conditions which the implicative operator in a logic system should be satisfied. Then the basic properties with different conditions and the characterizes of Implication Algebra are given. The iff conditions of a (basic) Implication Algebra to be regular are discussed. The relations between BasicImplication Algebra and Implication Algebra are gained when (X,-,0) is regular.From the view of lattices,some lattice properties of Implication Algebra and the conditions when a Implication Algebra is a lattice are found.2. Some other conditions which the implicative operator of a Implication Algebra should satisfied in a logic system are given. The relations between MV-Algebra and Distributive Implication Algebra,Implication Algebra with condition (S) are gained. Some equivalent theories about Implication Algebra and MV-Algebra are proved. The relations between Heyting Algebra and Implication Algebra with some conditions on a partial ordered set are discussed. Then some conditions when a Implication Algebra is a Boolean Algebra are given.3. The MP-filters and fuzzy filters of a Implication Algebra on a partial ordered set are studied with the condition given in chapter 2 which implicative operator should satisfy. The representation theories of MP-filter which is created by an non-empty set of a Implication Algebra on a partial ordered set with condition (C) are obtained at first. And it's proved that the set which contains all MP-filters of a Implication Algebra X,denoted by MF(X) = {F X F is a filter of X},is a distributive lattice and a completelattice also in the view of the concept of MP-filter. Then the fuzzy filter of a Implication Algebra is discussed,and the relations between MP-filter and fuzzy filter are obtained. Some characterizes when a fuzzy filter is a prime one are given. Some characterizes of a kind of Implication Algebra on a partial ordered set are gained by the use of MP-filter and fuzzy filter.