若干逻辑代数系统结构的研究

发布时间：2018-01-22 17:59

本文关键词： BCK/BCI代数伪BCK代数 NM代数(R_0代数) 伪NM代数正规R_0代数格蕴涵代数理想滤子　出处：《西安电子科技大学》2005年博士论文　论文类型：学位论文

【摘要】：逻辑代数是计算机科学、信息科学、控制论与人工智能等许多领域推理机制的代数基础。BCK／BCI代数是两类逻辑代数，BCI代数是BCK代数的推广。最近研究成果表明，偏序交换剩余整独异点(Pocrims)与具有条件(S)的BCK代数范畴同构，剩余格(Residuated lattices)与具有条件(S)的有界BCK格范畴同构。因此大部分关于逻辑的代数，如著名的MTL代数，BL代数，Heyting代数，MV代数(格蕴涵代数)，NM代数(R_0代数)，Boole代数等，都是BCK代数的自然扩张(即为BCK代数的子类)。由于p-半单BCI代数与Abel群范畴同构，因此Abel群是BCI代数的自然扩张。这些说明BCK／BCI代数是相当广泛的结构。因此，研究BCK／BCI代数就显得十分重要。近年来，由于来自理论与应用两个方面的推动，基于T模逻辑系统与对应的伪逻辑系统的研究成为逻辑领域中备受关注的热点之一，其中基于T模逻辑系统的研究先于逻辑代数，而伪逻辑代数的发展先于伪逻辑系统。NM代数(R_0代数)与格蕴涵代数皆是基于T模逻辑的代数。本文主要研究BCK／BCI代数及其扩张NM代数和格蕴涵代数的结构性质。具体工作如下： 1．引入一类新理想——BCI关联理想的概念，证明了它是BCK代数中关联理想概念在BCI代数中的自然推广。证明了BCI代数的一个非空子集是BCI关联理想当且仅当它既是BCI交换理想又是BCI正定关联理想，从而揭示了这三类理想之间的内在联系，，并将BCK代数中知名论断：BCK代数的一个非空子集是关联理想当且仅当它既是交换理想又是正定关联理想，推广到BCI代数上去。应用BCI关联理想完全刻画了关联BCI代数。引入FSI理想和FSC理想的概念，证明了BCI代数的一个Fuzzy子集是一个FSI理想当且仅当它是一个FSC理想和一个Fuzzy BCI正定关联理想． 2．构造了一类新的商BCK／BCI代数和一类新的有界商BCK代数，利用这种构造，各类型商BCK／BCI代数可以被相应的Fuzzy理想／滤子完全刻画，以往的商构造被Fuzzy理想／滤子刻画时只有充分条件而没有必要条件，因此新构造弥补了以往构造的不足，比以往的构造更加合理．证明了BCI代数的一个Fuzzy理想是闭的，当且仅当它是一个Fuzzy子代数．指出了在一些重要的BCI代数类中，任意Fuzzy理想必是闭的。 3．给出了BCK／BCI代数的Fuzzy极大理想的一个新定义，它比Hoo和Sessa
[Abstract]:Logical algebra is the algebraic basis of reasoning mechanism in many fields, such as computer science, information science, cybernetics and artificial intelligence. BCK / BCI algebra is two kinds of logic algebras. BCI algebras are a generalization of BCK algebras. Recent research results show that the partial order commutative residual integral unique points are isomorphic to the category of BCK algebras with conditional S). Residuated lattices) is isomorphic to the category of bounded BCK lattices with conditions.Therefore, most algebras about logic, such as the famous MTL algebra. BL algebras / Heyting algebras / MV-algebras (lattice implication algebras / NM algebras / R _ S _ 0 algebras / Boole algebras etc.). Both are natural extensions of BCK algebras (that is, subclasses of BCK algebras) because p-semisimple BCI algebras and Abel group categories are isomorphic. Therefore, Abel groups are natural extensions of BCI algebras. These show that BCK/BCI algebras are quite extensive structures. Therefore, it is very important to study BCK/BCI algebras. In recent years, due to the promotion of theory and application, the research of T-module logic system and corresponding pseudo-logic system has become one of the hot topics in the field of logic. The research of T module logic system is prior to logic algebra, and the development of pseudo logic algebra is prior to pseudo logic system. NM algebra and lattice implication algebra are all algebras based on T module logic. In this paper, we study the structural properties of BCK/BCI algebras and their extended NM algebras and lattice implication algebras. 1. The concept of BCI associative ideal is introduced. It is proved that it is a natural generalization of the concept of associative ideals in BCK algebras in BCI algebras. It is proved that a nonempty subset of BCI algebras is a BCI associated ideal if and only if it is both a BCI commutative ideal and a B. CI positive definite correlation ideal. In this paper, the inner relation between these three kinds of ideals is revealed, and a non-empty subset of the BCK algebra is an associative ideal if and only if it is both a commutative ideal and a positive definite associative ideal. In this paper, we generalize to BCI algebras. We use BCI associative ideals to characterize associative BCI algebras completely. The concepts of FSI ideals and FSC ideals are introduced. It is proved that a Fuzzy subset of BCI algebra is a FSI ideal if and only if it is a FSC ideal and a Fuzzy BCI positive definite associative ideal. 2. A new class of quotient BCK/BCI algebras and a new class of bounded quotient BCK algebras are constructed. All types of quotient BCK/BCI algebras can be completely characterized by the corresponding Fuzzy ideals / filters. In the past quotient constructions were characterized by Fuzzy ideals / filters with only sufficient conditions and no necessary conditions. Therefore, the new structure makes up for the deficiency of the previous structure and is more reasonable than the previous one. It is proved that a Fuzzy ideal of BCI algebra is closed. If and only if it is a Fuzzy subalgebra, it is pointed out that in some important classes of BCI algebras, any Fuzzy ideal must be closed. 3. A new definition of Fuzzy maximal ideal of BCK/BCI algebra is given, which is better than that of Hoo and Sessa.
【学位授予单位】：西安电子科技大学
【学位级别】：博士
【学位授予年份】：2005
【分类号】：O141.1

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