代数系统的可嵌入性
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摘要
自1965年L.A.Zadeh提出模糊集的概念以来，关于模糊系统的研究得到了迅猛的发展，模糊控制技术被广泛应用于工业控制与家电产品的制造中，并取得了令人瞩目的成功。然而，模糊逻辑缺乏深入的理论研究，特别是模糊推理这一模糊控制原理的核心部分，缺乏严格的逻辑基础。其中的合成推理方法存在着某种缺陷，使得模糊推理与模糊逻辑没有很好地结合起来，这导致了实质在于质疑模糊推理方法的理论基础的一场论战。为了促进模糊逻辑与模糊推理的融合与发展，王国俊教授作了大量深入细致的工作，并取得了一系列有意义的成果。这些研究成果已使模糊推理缺乏逻辑基础的状况得到了较大的改善。在完全解决模糊推理的逻辑基础问题中，形式演绎系统的完备性是非经典逻辑的主要研究方向之一。对于一个形式系统而言，完备性是至关重要的逻辑性质，它反映了该系统语法与语义的和谐性。正是为了追求这种和谐性，非经典逻辑领域的许多学者进行了大量的研究成果，取得了大批重要的理论成果。而在完备性问题研究中，可嵌入性是一个关键的问题，无论是证明Lukasiewicz的完备性还是系统(?)的完备性问题，可嵌入性发挥了至关重要的作用。在可嵌入性的保证下，当一个公式对所有的某种线性代数系统是重言式时，其必定对所有的同种代数系统是重言式。而线性代数系统讨论起来就方便多了。从而研究代数系统的可嵌入性是非常有意义的。本文的目的正是研究几类重要的代数系统的可嵌入性问题。
本文的主要内容如下：
第一章：讨论了FI代数及其MP滤子的性质；给出了LFI代数及CLFI代数的定义；讨论了LFI代数的性质；在此基础上，给出了LFI代数可嵌入于全序FI代数乘积的充要条件。事实上，关于LFI代数的这一方法完全可“平移”到讨论剩余格的可嵌入性问题上。进而还指出了具有可嵌入性的LFI代数的性质。
第二章：讨论了满足(*)式的剩余格的一些性质；找到了剩余格可嵌入于全序剩余格乘积的充要条件；利用这个结果，得到了几类重要代数系统像R_0代数、MV代数、BL代数、WNM代数、蕴涵格等等具有可嵌入性；还讨论了LFI代数与剩余格的关系，给出了CLFI代数成为剩余格的充要条件。
第三章：从多方面研究了MTL代数的MP滤子，给出MTL代数极大MP滤子的定义；找出了是MTL代数极大MP滤子的等价条件；对MTL代数的子代数给予刻划；还讨论了局部有限MTL代数；研究了MTL代数的布尔滤子；证明了当F是MTL代数M的布尔滤子时，M／～F是布尔代数；还讨论了局部MTL代数。因为MTL代数是一类比较基本的代数系统，像R_0代数、MV代数、WNM代数等等都是MTL代数，所以这些结果在上述代数系统中也是成立的。
第四章：证明了当两个阿基米德卜范的减生成子具有某种关系时，这两个阿
基米德t－范是同构的；这个工作使得仅需判断两个阿基米德t－范的减生成子在0
处的值，便可以搞清楚它们是否同构，这在很大程度上简化了Hdjek的工作．指
出了由严格的仁范定义的蕴涵算子方（＄叨在叮0）处不连续；并给出了什么样的
蕴涵算子定义的t－范是阿基米德的．
Since L.A.Zadeh proposed the concept of fuzzy set in 1965. the study on fuzzy system has made rapid development. Fuzzy control technology was widely applied in industry control and the production of family electrical equipments, which made enormous success. But its basic theory-fuzzy reasoning, still hasn't a dependable logical base. In order to promote the combination and development of fuzzy logic and fuzzy reasoning, Professor Wang Guo-jun has done a great deal of work and obtained a series of meaningful results. About this, the completeness of the formal deductive systems is one of the main branches. For a formal deductive system, the completeness is a crucial logical property. It embodies the harmony between syntax and semantic . However, the embeddability plays an important role in proving the completeness of the formal deductive systems. When an algebra system has embeddability. a formula is a tautology for each linearly ordered algebra if and only if it is a tautology for each algebra. While, a linearly ordered algebra is convenient to discuss. So it is very significant to study the embeddability of algebra systems. This paper mainly discusses the embeddability of several kinds of important algebra systems .
The main content of the present paper is as follow:
In the first part, this paper gives the definition of LFI algebras and CLFI algebras; discusses the properties of FI algebras, theirs MP filters. LFI algebras and LFI algebras with embeddability. On the basis of this, this paper gives a necessary and sufficient condition for an LFI algebra being embedded into the product of a system of linearly ordered FI algebras. In fact, this method wholly suits residuated lattices.
The second part discusses the properties of residuated lattices satisfying ; gives a necessary and sufficient condition w.r.t. residuated lattices. By means of it, this paper obtains several kinds of important algebra systems such as R0 algebras, MV algebras, BL algebras, and WNM algebras have embeddability; discusses the relation between LFI algebras and residuated lattices; gives a necessary and sufficient condition for CLFI algebras becoming residuated lattices.
In the third part, this paper studies the MP filters of MTL algebras from many respects. It gives the definition of maximal MP filters, provides the equivalent condition of maximal MP filters, depicts the structure of the subalgebras of MTL algebras, discusses locally finite MTL algebras; studies the Boolean filters of MTL algebras, proves if F is a Boolean filters of M which is an MTL algebra, then M/ -F is a Boolean algebra and discusses local MTL algebras. Because MTL algebras are a kind of basic algebra
systems,these results are true in other algebra systems such as R0 algebras, MV algebras, BL algebras and WNM algebras
Finally, the paper proves that two Archimedean t-norms are isomophic when their corresponding decreasing generators have certain relation, this work enables us to easily judge if two Archimedean are isomophic. This simplifies Hajek's work to a great extent. It points out that the implication operator I(x, y) defined by strict t-norms is not continuous at (0,0) and discusses under what conditions a t-norm induced by an implication operator is Archimedean.
本文的主要内容如下：
第一章：讨论了FI代数及其MP滤子的性质；给出了LFI代数及CLFI代数的定义；讨论了LFI代数的性质；在此基础上，给出了LFI代数可嵌入于全序FI代数乘积的充要条件。事实上，关于LFI代数的这一方法完全可“平移”到讨论剩余格的可嵌入性问题上。进而还指出了具有可嵌入性的LFI代数的性质。
第二章：讨论了满足(*)式的剩余格的一些性质；找到了剩余格可嵌入于全序剩余格乘积的充要条件；利用这个结果，得到了几类重要代数系统像R_0代数、MV代数、BL代数、WNM代数、蕴涵格等等具有可嵌入性；还讨论了LFI代数与剩余格的关系，给出了CLFI代数成为剩余格的充要条件。
第三章：从多方面研究了MTL代数的MP滤子，给出MTL代数极大MP滤子的定义；找出了是MTL代数极大MP滤子的等价条件；对MTL代数的子代数给予刻划；还讨论了局部有限MTL代数；研究了MTL代数的布尔滤子；证明了当F是MTL代数M的布尔滤子时，M／～F是布尔代数；还讨论了局部MTL代数。因为MTL代数是一类比较基本的代数系统，像R_0代数、MV代数、WNM代数等等都是MTL代数，所以这些结果在上述代数系统中也是成立的。
第四章：证明了当两个阿基米德卜范的减生成子具有某种关系时，这两个阿
基米德t－范是同构的；这个工作使得仅需判断两个阿基米德t－范的减生成子在0
处的值，便可以搞清楚它们是否同构，这在很大程度上简化了Hdjek的工作．指
出了由严格的仁范定义的蕴涵算子方（＄叨在叮0）处不连续；并给出了什么样的
蕴涵算子定义的t－范是阿基米德的．
Since L.A.Zadeh proposed the concept of fuzzy set in 1965. the study on fuzzy system has made rapid development. Fuzzy control technology was widely applied in industry control and the production of family electrical equipments, which made enormous success. But its basic theory-fuzzy reasoning, still hasn't a dependable logical base. In order to promote the combination and development of fuzzy logic and fuzzy reasoning, Professor Wang Guo-jun has done a great deal of work and obtained a series of meaningful results. About this, the completeness of the formal deductive systems is one of the main branches. For a formal deductive system, the completeness is a crucial logical property. It embodies the harmony between syntax and semantic . However, the embeddability plays an important role in proving the completeness of the formal deductive systems. When an algebra system has embeddability. a formula is a tautology for each linearly ordered algebra if and only if it is a tautology for each algebra. While, a linearly ordered algebra is convenient to discuss. So it is very significant to study the embeddability of algebra systems. This paper mainly discusses the embeddability of several kinds of important algebra systems .
The main content of the present paper is as follow:
In the first part, this paper gives the definition of LFI algebras and CLFI algebras; discusses the properties of FI algebras, theirs MP filters. LFI algebras and LFI algebras with embeddability. On the basis of this, this paper gives a necessary and sufficient condition for an LFI algebra being embedded into the product of a system of linearly ordered FI algebras. In fact, this method wholly suits residuated lattices.
The second part discusses the properties of residuated lattices satisfying ; gives a necessary and sufficient condition w.r.t. residuated lattices. By means of it, this paper obtains several kinds of important algebra systems such as R0 algebras, MV algebras, BL algebras, and WNM algebras have embeddability; discusses the relation between LFI algebras and residuated lattices; gives a necessary and sufficient condition for CLFI algebras becoming residuated lattices.
In the third part, this paper studies the MP filters of MTL algebras from many respects. It gives the definition of maximal MP filters, provides the equivalent condition of maximal MP filters, depicts the structure of the subalgebras of MTL algebras, discusses locally finite MTL algebras; studies the Boolean filters of MTL algebras, proves if F is a Boolean filters of M which is an MTL algebra, then M/ -F is a Boolean algebra and discusses local MTL algebras. Because MTL algebras are a kind of basic algebra
systems,these results are true in other algebra systems such as R0 algebras, MV algebras, BL algebras and WNM algebras
Finally, the paper proves that two Archimedean t-norms are isomophic when their corresponding decreasing generators have certain relation, this work enables us to easily judge if two Archimedean are isomophic. This simplifies Hajek's work to a great extent. It points out that the implication operator I(x, y) defined by strict t-norms is not continuous at (0,0) and discusses under what conditions a t-norm induced by an implication operator is Archimedean.
引文
[1] L.A. Zadeh, Fuzzy sets, Information and Control, 1965, 8(3), 338-353.
[2] J.A. Goguen, The logic of inexact concepts, synthese. 1968, 9. 19. 325-373.
[3] L. Bolc, P. Borowik, 《Many-Valued Logic(Ⅰ)》 , Springer-Verlag, 1992.
[4] J. Pavelka, On fuzzy logic(Ⅰ,Ⅱ,Ⅲ), Z. für Mathematik Logicu Gruudlagen d Mathematic, 1979, 25, 45-52, 119-134, 447-464.
[5] 吴望名，关于模糊逻辑的一场争论，模糊系统与数学，1995，9(2)，1-9．
[6] 王国俊，模糊逻辑与模糊推理，全国第七界多值逻辑与模糊逻辑学术会议论文集，西安，1996，82-96．
[7] 王国俊，一类代数上的．逻辑学(Ⅰ，Ⅱ)．陕西师范大学学报(自然科学版)，1997，25(1)，1-8，25(3)，1-8．
[8] 王国俊，Fuzzy命题演算的一种形式演绎系统，科学通报，1997，42(10)，1041-1045．
[9] G.J. Wang, On the logic foundations of fuzzy modus ponens and fuzzy modus tollens, J. Fuzzy Mathematics, 1997, 5(1), 229-250.
[10] G. J. Wang, On the logic foundation of fuzzy reasoning, Lecture Notes in Fuzzy Mathematics and Computer Science, Omaha(USA), Creighton University, 1997, 4, 1-48.
[11] 王国俊，修正的Kleene系统中的∑-(α-重言式)理论，中国科学(E辑)，1998，18(2)，146-152．
[12] 王国俊，模糊推理的全蕴涵三Ⅰ算法，中国科学(E辑)，1999，29(1)，43-53．
[13] 王国俊，模糊推理的一个新方法，模糊系统与数学，1999，13(3)，1-10．
[14] 王国俊，论聚合与推理的顺序，陕西师范大学学报(自)，1999，27(1)，1-5．
[15] Wang Guojun,Triple I method and interval valued fuzzy reasoning, Science in China(Ser.E),2000,43(3),242-253.
[16] 王国俊，《非经典数理逻辑与近似推理》，北京，科学出版社，2000．
[17] C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms, ibid., 1959, 93(1), 74-80.
[18] C. C. Chang, Algebraic analysis of many-valued logic, Trans. A. M. S., 1958,88, 467-490.
[19] P. Hájek, 《Metamathematics of Fuzzy Logic》, Kluwer Academic Publisher, 1998.
[20] 刘军，基于格蕴涵代数的格值逻辑系统及格值归结原理的研究，西南交通大学博士学位论文，1999．
[21] Xu Yang，et al.L-valued propositional logic L_(vpl),Inform.Sci.,1999.144,205-235.
[22] 裴道武，王国俊，形式系统的完备性及其应用，中国科学(E)，2002： 32(1)，56-64．
[23] 吴望名，Fuzzy蕴涵代数，模糊系统与数学，1990：4(1)，56-64．
[24] 裴道武，R_0代数中的MP滤子与同余关系，模糊系统与数学，2000，14，22-25．
[25] 裴道武，四川大学博士学位论文，2001．
[26] 裴道武，剩余格与正则剩余格的几个特征定理，数学学报，2000，45(2)， 271-278．
[27] 傅丽，次BL代数的推理系统，陕西师范大学学报，2002，30(1)，17-21．
[28] 王国俊，MV代数，BL代数，R_0代数与多值逻辑，模糊系统与数学，2002，16(2)，1-15．
[29] F.Esteva, L.Godo,Monoidal t-norm based logic:towards a logic for left continuous t-norm,Fuzzy Sets and Systems,2001,124(3),271-288.
[30] 王国俊，蕴涵格及其Fuzzy拓扑表现定理，数学学报，1999，41(1)．
[31] 袁和军，陕西师范大学硕士学位论文，2002．
[32] 任芳，陕西师范大学硕士学位论文，2001．
[33] C.H.Ling,Representation of associative functions,Publ.Math.Debrecen, 1965(12),189-212.
[34] 李中夫，刘应明，用单个一元单调函数与加法近似表示一类可结合函数，中国科学，A辑，1993，23，1246-1253．
[35] E.Turunen,《Mathematics Behind Fuzzy logic》,Physica-Verlag,1999.
[36] E.Turunen,Boolean deductive systems of BL-algebras,Arch.Math.Logic, 2001,40,467-473.
[37] 刘军，徐扬，格蕴涵代数与结构，科学通报，1997，42(10)，1049-1051．
[38] 孟彪龙，格蕴涵代数的素滤子，西北大学学报，1998，28(3)，189-192．
[39] 吴望名，《模糊推理的原理和方法》，贵阳，贵州科技出版社，1994．
[40] 胡长流，宋振明，《格论基础》，开封，河南大学出版社，1990．
[41] 胡淑礼，《模糊数学及其应用》，成都，四川大学出版社，1994．
[42] 张文修，梁怡，《不确定性推理原理》，西安，西安交通大学出版社，1996．
[43] 刘连珍，王国俊，Fuzzy蕴涵代数与MV代数，模糊系统与数学，1998，12(1)．20-25．
[2] J.A. Goguen, The logic of inexact concepts, synthese. 1968, 9. 19. 325-373.
[3] L. Bolc, P. Borowik, 《Many-Valued Logic(Ⅰ)》 , Springer-Verlag, 1992.
[4] J. Pavelka, On fuzzy logic(Ⅰ,Ⅱ,Ⅲ), Z. für Mathematik Logicu Gruudlagen d Mathematic, 1979, 25, 45-52, 119-134, 447-464.
[5] 吴望名，关于模糊逻辑的一场争论，模糊系统与数学，1995，9(2)，1-9．
[6] 王国俊，模糊逻辑与模糊推理，全国第七界多值逻辑与模糊逻辑学术会议论文集，西安，1996，82-96．
[7] 王国俊，一类代数上的．逻辑学(Ⅰ，Ⅱ)．陕西师范大学学报(自然科学版)，1997，25(1)，1-8，25(3)，1-8．
[8] 王国俊，Fuzzy命题演算的一种形式演绎系统，科学通报，1997，42(10)，1041-1045．
[9] G.J. Wang, On the logic foundations of fuzzy modus ponens and fuzzy modus tollens, J. Fuzzy Mathematics, 1997, 5(1), 229-250.
[10] G. J. Wang, On the logic foundation of fuzzy reasoning, Lecture Notes in Fuzzy Mathematics and Computer Science, Omaha(USA), Creighton University, 1997, 4, 1-48.
[11] 王国俊，修正的Kleene系统中的∑-(α-重言式)理论，中国科学(E辑)，1998，18(2)，146-152．
[12] 王国俊，模糊推理的全蕴涵三Ⅰ算法，中国科学(E辑)，1999，29(1)，43-53．
[13] 王国俊，模糊推理的一个新方法，模糊系统与数学，1999，13(3)，1-10．
[14] 王国俊，论聚合与推理的顺序，陕西师范大学学报(自)，1999，27(1)，1-5．
[15] Wang Guojun,Triple I method and interval valued fuzzy reasoning, Science in China(Ser.E),2000,43(3),242-253.
[16] 王国俊，《非经典数理逻辑与近似推理》，北京，科学出版社，2000．
[17] C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms, ibid., 1959, 93(1), 74-80.
[18] C. C. Chang, Algebraic analysis of many-valued logic, Trans. A. M. S., 1958,88, 467-490.
[19] P. Hájek, 《Metamathematics of Fuzzy Logic》, Kluwer Academic Publisher, 1998.
[20] 刘军，基于格蕴涵代数的格值逻辑系统及格值归结原理的研究，西南交通大学博士学位论文，1999．
[21] Xu Yang，et al.L-valued propositional logic L_(vpl),Inform.Sci.,1999.144,205-235.
[22] 裴道武，王国俊，形式系统的完备性及其应用，中国科学(E)，2002： 32(1)，56-64．
[23] 吴望名，Fuzzy蕴涵代数，模糊系统与数学，1990：4(1)，56-64．
[24] 裴道武，R_0代数中的MP滤子与同余关系，模糊系统与数学，2000，14，22-25．
[25] 裴道武，四川大学博士学位论文，2001．
[26] 裴道武，剩余格与正则剩余格的几个特征定理，数学学报，2000，45(2)， 271-278．
[27] 傅丽，次BL代数的推理系统，陕西师范大学学报，2002，30(1)，17-21．
[28] 王国俊，MV代数，BL代数，R_0代数与多值逻辑，模糊系统与数学，2002，16(2)，1-15．
[29] F.Esteva, L.Godo,Monoidal t-norm based logic:towards a logic for left continuous t-norm,Fuzzy Sets and Systems,2001,124(3),271-288.
[30] 王国俊，蕴涵格及其Fuzzy拓扑表现定理，数学学报，1999，41(1)．
[31] 袁和军，陕西师范大学硕士学位论文，2002．
[32] 任芳，陕西师范大学硕士学位论文，2001．
[33] C.H.Ling,Representation of associative functions,Publ.Math.Debrecen, 1965(12),189-212.
[34] 李中夫，刘应明，用单个一元单调函数与加法近似表示一类可结合函数，中国科学，A辑，1993，23，1246-1253．
[35] E.Turunen,《Mathematics Behind Fuzzy logic》,Physica-Verlag,1999.
[36] E.Turunen,Boolean deductive systems of BL-algebras,Arch.Math.Logic, 2001,40,467-473.
[37] 刘军，徐扬，格蕴涵代数与结构，科学通报，1997，42(10)，1049-1051．
[38] 孟彪龙，格蕴涵代数的素滤子，西北大学学报，1998，28(3)，189-192．
[39] 吴望名，《模糊推理的原理和方法》，贵阳，贵州科技出版社，1994．
[40] 胡长流，宋振明，《格论基础》，开封，河南大学出版社，1990．
[41] 胡淑礼，《模糊数学及其应用》，成都，四川大学出版社，1994．
[42] 张文修，梁怡，《不确定性推理原理》，西安，西安交通大学出版社，1996．
[43] 刘连珍，王国俊，Fuzzy蕴涵代数与MV代数，模糊系统与数学，1998，12(1)．20-25．