Study On The Lattice Implication Algebraic Equations

Lattice implication algebra is a kind of logic algebra, and it is the basis for researching on lattice-valued logic theory; the purpose of researching on lattice-valued logic is to provide a kind of logical foundation for uncertainty reasoning and automated reasoning. We often encounter the problems of solvability of equations on lattice implication algebra in the process of researching on uncertainty reasoning and automated reasoning. At the same time, with developing of both theory and application of lattice-valued logic based on lattice implication algebras, it is inevitable that the solvability of finite or infinite lattice-valued equations on lattice implication algebra will arise.The author researched into the algebraic equations on lattice implication algebra. Following are the main works contained in this thesis:1. Studied the relations between lattice implication algebra and Brouwerian lattice. Proved that all LI -ideals of a lattice H implication form a complete Brouwerian lattice, and pointed out that if L is a complete lattice implication algebra, then (L, ∨, ∧) is a Brouwerian lattice.2. Lattice implication algebraic equation is defined, several kinds of simple lattice implication algebraic equations are discussed, corresponding conditions about solvability of equations are presented, and least and greatest solutions of equations are discussed. When domain is a complete lattice implication algebra, the solution sets of equations are characterized. In the end, some properties of solution sets are discussed.3. When domain is lattice implication algebra, on the one hand, for the " ∧ - ∧ " finite lattice implication algebraic equations, the notion of least relativepseudo-complement lattice is introduced, it is called LRPC lattice for short; and the least solution is presented when L is a complete lattice implication algebra. On the other hand, for the "∧→ " finite lattice implication algebraic equations, presented the least solution. For the two kinds of equations, respectively, conditions about solvability of equations are presented, and proved that both their solution sets form semi-lattice. In the case of several special conditions, constructed all the extremum solutions of equations, the number of extremum solutions are formulated. Further, when L is a complete lattice implication algebra, characterized the solution set of equations. Finally, some properties of solution set are discussed.