本文關(guān)鍵詞： BCK/BCI代數(shù) 偽BCK代數(shù) NM代數(shù)(R_0代數(shù)) 偽NM代數(shù) 正規(guī)R_0代數(shù) 格蘊(yùn)涵代數(shù) 理想 濾子　出處：《西安電子科技大學(xué)》2005年博士論文　論文類型：學(xué)位論文

【摘要】：邏輯代數(shù)是計(jì)算機(jī)科學(xué)、信息科學(xué)、控制論與人工智能等許多領(lǐng)域推理機(jī)制的代數(shù)基礎(chǔ)。BCK／BCI代數(shù)是兩類邏輯代數(shù)，BCI代數(shù)是BCK代數(shù)的推廣。最近研究成果表明，偏序交換剩余整獨(dú)異點(diǎn)(Pocrims)與具有條件(S)的BCK代數(shù)范疇同構(gòu)，剩余格(Residuated lattices)與具有條件(S)的有界BCK格范疇同構(gòu)。因此大部分關(guān)于邏輯的代數(shù)，如著名的MTL代數(shù)，BL代數(shù)，Heyting代數(shù)，MV代數(shù)(格蘊(yùn)涵代數(shù))，NM代數(shù)(R_0代數(shù))，Boole代數(shù)等，都是BCK代數(shù)的自然擴(kuò)張(即為BCK代數(shù)的子類)。由于p-半單BCI代數(shù)與Abel群范疇同構(gòu)，因此Abel群是BCI代數(shù)的自然擴(kuò)張。這些說(shuō)明BCK／BCI代數(shù)是相當(dāng)廣泛的結(jié)構(gòu)。因此，研究BCK／BCI代數(shù)就顯得十分重要。 近年來(lái)，由于來(lái)自理論與應(yīng)用兩個(gè)方面的推動(dòng)，基于T模邏輯系統(tǒng)與對(duì)應(yīng)的偽邏輯系統(tǒng)的研究成為邏輯領(lǐng)域中備受關(guān)注的熱點(diǎn)之一，其中基于T模邏輯系統(tǒng)的研究先于邏輯代數(shù)，而偽邏輯代數(shù)的發(fā)展先于偽邏輯系統(tǒng)。NM代數(shù)(R_0代數(shù))與格蘊(yùn)涵代數(shù)皆是基于T模邏輯的代數(shù)。 本文主要研究BCK／BCI代數(shù)及其擴(kuò)張NM代數(shù)和格蘊(yùn)涵代數(shù)的結(jié)構(gòu)性質(zhì)。具體工作如下： 1．引入一類新理想——BCI關(guān)聯(lián)理想的概念，證明了它是BCK代數(shù)中關(guān)聯(lián)理想概念在BCI代數(shù)中的自然推廣。證明了BCI代數(shù)的一個(gè)非空子集是BCI關(guān)聯(lián)理想當(dāng)且僅當(dāng)它既是BCI交換理想又是BCI正定關(guān)聯(lián)理想，從而揭示了這三類理想之間的內(nèi)在聯(lián)系，，并將BCK代數(shù)中知名論斷：BCK代數(shù)的一個(gè)非空子集是關(guān)聯(lián)理想當(dāng)且僅當(dāng)它既是交換理想又是正定關(guān)聯(lián)理想，推廣到BCI代數(shù)上去。應(yīng)用BCI關(guān)聯(lián)理想完全刻畫了關(guān)聯(lián)BCI代數(shù)。引入FSI理想和FSC理想的概念，證明了BCI代數(shù)的一個(gè)Fuzzy子集是一個(gè)FSI理想當(dāng)且僅當(dāng)它是一個(gè)FSC理想和一個(gè)Fuzzy BCI正定關(guān)聯(lián)理想． 2．構(gòu)造了一類新的商BCK／BCI代數(shù)和一類新的有界商BCK代數(shù)，利用這種構(gòu)造，各類型商BCK／BCI代數(shù)可以被相應(yīng)的Fuzzy理想／濾子完全刻畫，以往的商構(gòu)造被Fuzzy理想／濾子刻畫時(shí)只有充分條件而沒(méi)有必要條件，因此新構(gòu)造彌補(bǔ)了以往構(gòu)造的不足，比以往的構(gòu)造更加合理．證明了BCI代數(shù)的一個(gè)Fuzzy理想是閉的，當(dāng)且僅當(dāng)它是一個(gè)Fuzzy子代數(shù)．指出了在一些重要的BCI代數(shù)類中，任意Fuzzy理想必是閉的。 3．給出了BCK／BCI代數(shù)的Fuzzy極大理想的一個(gè)新定義，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.
【學(xué)位授予單位】：西安電子科技大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2005
【分類號(hào)】：O141.1

【引證文獻(xiàn)】

相關(guān)期刊論文 前1條

1 宮洪娟;張小紅;;psBCK-代數(shù)的Boolean濾子與psMV-濾子[J];寧波大學(xué)學(xué)報(bào)(理工版);2011年01期

相關(guān)碩士學(xué)位論文 前1條

1 杜紹坤;格蘊(yùn)涵代數(shù)及其與相關(guān)邏輯代數(shù)的關(guān)系研究[D];西南交通大學(xué);2011年

本文編號(hào)：1455365