【作者】 潘小东；

【导师】 徐扬；

【作者基本信息】 西南交通大学 ， 应用数学， 2005， 硕士

【摘要】 格蕴涵代数是一种逻辑代数,它是研究格值逻辑理论的一种基础。研究格值逻辑理论的目的是为了给不确定性推理和自动推理提供一种逻辑理论基础。在对不确定性推理和自动推理进行研究的过程中往往会遇到格蕴涵代数中方程的可解性问题。同时,随着基于格蕴涵代数的格值逻辑在理论和应用两个方面的进一步发展,它必然也会涉及到论域在格蕴涵代数上的有限或无限格值方程的可解性问题。 基于上述的研究背景,本文对格蕴涵代数中的格蕴涵代数方程进行了研究,主要做了下面几个方面的工作: 1.研究了格蕴涵代数与Brouwerian格之间的一些联系。证明了格H蕴涵代数的所有LI-理想构成一个完备的Brouwerian格。并且指出:若L是一个完备格蕴涵代数,则(L,∨,∧)是一个Brouwerian格。 2.对格蕴涵代数方程的概念进行了定义,讨论了几类结构简单的格蕴涵代数方程,给出了方程的可解性判别条件,并且讨论了方程的最小(大)解。当论域是完备格蕴涵代数时,对方程的解集进行了刻画;最后,讨论了解集的若干性质。 3.在论域是格蕴涵代数的情况之下,一方面,对于“∧-∨”型有限格蕴涵代数方程,引入了最小相对伪补格的概念,简记为:LRPC格;在此基础之上,当L是完备格蕴涵代数时,给出了方程的最小解。另一方面,对于“∧-→”型有限格蕴涵代数方程,给出了方程的最小解。对这两类方程,分别给出了方程的可解性判别条件,证明了它们的解集均构成一个半格。在几种特殊情况之下,构造出了方程的所有极值解,并给出了极值解的个数公式。进一步,在论域是完备格蕴涵代数时,刻画出了方程的解集。最后讨论了解集的若干性质。更多还原

【Abstract】 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.更多还原