关于格蕴涵代数性质的研究
摘要
非经典逻辑是人工智能领域中十分活跃的研究方向，是不确定性推理的理论基础。在非经典逻辑的研究中，格值逻辑的研究具有重要而广泛的意义。格蕴涵代数是徐扬教授为研究格值逻辑把格与蕴涵代数相结合提出的一个代数系统。本文将在格蕴涵代数已有性质的基础上，进一步讨论格蕴涵代数的性质及结构。具体作了以下三方面的工作：
1、在有界分配格的素理想空间上给出了格蕴涵代数的一个表示，并借助集合的无限运算系统地讨论了表示的性质，得出了格蕴涵代数中任意子集无限运算的若干性质。
2、对格蕴涵代数中由一子集生成的格蕴涵子代数的结构及格蕴涵代数的直积进行了研究，给出了一格蕴涵代数可分解成一族格蕴涵代数直积的必要条件，证明了格H蕴涵代数是一有单位元的布尔环。
3、给出了格蕴涵代数中O-理想的定义并讨论了其性质，给出了格蕴涵代数中集合S上超滤子的定义并与集合代数上的超滤子作了比较，研究了两种超滤子之间的关系。
Non-classical logic is an active research direction in the field of artificial intelligence and a logic foundation for uncertainty reasoning. In the study of non-classical logic, lattice-valued logic system is of extensive significance. Lattice implication algebra is an algebraic system combinating lattice with implication algebra. It is first defined by professor Xu Yang in order to study lattice-valued logic. There are many research papers about lattice implication algebra and related logic. In this paper, the proper and structure of lattice implication algebra is further studied. The following works have been done:
1. On the prime ideal space of distributive lattice with bounds, a representation of lattice implication algebra was given. And with the help of the limit of set, the properties of this representation were discussed. Finally, some of properties of any subset's limit of lattice implication algebra have been proved.
2.The structure of the lattice implication sub-algebra which generated from a subset, and the product of lattice implication algebra were studied. The necessary condition was given of a lattice implication algebra which can be decomposed into lattice implication algebra's product. It is proved that lattice H implication algebra is a Boole ring with a unit.
3.The concepts of O-ideal was proposed and its property was discussed. Ultra-filter on a set S of lattice implication algebra was defined and was compared with the ultra-filter on a set of lattice, the relationship between this two ultra-filter was studied.
1、在有界分配格的素理想空间上给出了格蕴涵代数的一个表示，并借助集合的无限运算系统地讨论了表示的性质，得出了格蕴涵代数中任意子集无限运算的若干性质。
2、对格蕴涵代数中由一子集生成的格蕴涵子代数的结构及格蕴涵代数的直积进行了研究，给出了一格蕴涵代数可分解成一族格蕴涵代数直积的必要条件，证明了格H蕴涵代数是一有单位元的布尔环。
3、给出了格蕴涵代数中O-理想的定义并讨论了其性质，给出了格蕴涵代数中集合S上超滤子的定义并与集合代数上的超滤子作了比较，研究了两种超滤子之间的关系。
Non-classical logic is an active research direction in the field of artificial intelligence and a logic foundation for uncertainty reasoning. In the study of non-classical logic, lattice-valued logic system is of extensive significance. Lattice implication algebra is an algebraic system combinating lattice with implication algebra. It is first defined by professor Xu Yang in order to study lattice-valued logic. There are many research papers about lattice implication algebra and related logic. In this paper, the proper and structure of lattice implication algebra is further studied. The following works have been done:
1. On the prime ideal space of distributive lattice with bounds, a representation of lattice implication algebra was given. And with the help of the limit of set, the properties of this representation were discussed. Finally, some of properties of any subset's limit of lattice implication algebra have been proved.
2.The structure of the lattice implication sub-algebra which generated from a subset, and the product of lattice implication algebra were studied. The necessary condition was given of a lattice implication algebra which can be decomposed into lattice implication algebra's product. It is proved that lattice H implication algebra is a Boole ring with a unit.
3.The concepts of O-ideal was proposed and its property was discussed. Ultra-filter on a set S of lattice implication algebra was defined and was compared with the ultra-filter on a set of lattice, the relationship between this two ultra-filter was studied.
引文
[1] 尹朝庆，尹皓，人工智能与专家系统，中国水利水电出版社，(2002)1-6．
[2] 宋振明，基于格蕴涵代数的智能控制中不确定性推理的研究，西南交通大学博士论文，(1998)14-15．
[3] J.Lukasiewicz, On 3-valued logic, Ruch Filozoficzny, 5(1920)169-170.
[4] 秦克云，基于格蕴涵代数的格值逻辑系统及应用的研究，西南交通大学博士论文，(1996)2-6．
[5] D.A.Bochvav, On a 3-valued logic calculus and its application to the analysis of contradictions matematiceskij sbornik, 4(1939),287-308.
[6] 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，西南交通大学博士论文，(1998)3-8．
[7] E.Post,Introduction to a general theory of elementary propositions,American Journal of Mathematics, 43(1921)163-185.Reprinted in J.Van Heijenoort(ed.),From Frege to Godel:A source book in Mathematical Logic 1879-1931 ,Cambridge,Mass, 1967.
[8] Leonard Bolc, Piotr Borowik, Many-valued Logics, Springer-Verlag, Berlin Heidelberg New York, 1992.
[9] L.A.Zadeh, Fuzzy sets, Inf. Control, 8(1965),338-353.
[10] G.Birkhoff, Lattice Theory, 3rd edition, American Mathematical Society, Proridence, R.L,1967.
[11] J.A.Goguen,The logic of inexact concepts, Synthese, 19(1969),325-373.
[12] J.Lukasiewicz, Selected Works-Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam/PWN,Warzawa, 1971.
[13] Saunders Mac Lane, Categories for the Working Mathematician, Springer Verlag Berlin, 1971.
[14] C.C.Chang, Keisler H.J.,Model Theory, North-Holland Publishing Company, Amsterdam, 1973.
[15] Zhang Jinwen,A unified treatment of fuzzy set theory and boolean valued set theory-fuzzy set structures and nomal fuzzy set structure, Journal of Mathematical Analysis and Applications,76(1980),297-301.
[16] C.C.Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc. 88(1958)195-234.
[17] P. Mangani,On certain algebras related to many-valued logics, Boll.V.M.I 8(4)(1973)68-78.
[18] X.H.Liu, H.Xiao, Operator fuzzy logic and fuzzy resolution, Proc. Of the 5th IEEE Inter. Symp. on Multiple-valued Logic, ISMUL 85, Kingstion, Canada, (1985)68-75.
[19] K.Lano,Fuzzy sets and residuated logic, Fuzzy Sets and System 47(2)(1969)203-220.
[20] J.Pavelka, On fuzzy logic Ⅰ:Many-valued rules of inference, Ⅱ: Enriched residuated lattices and semantics of prepositional calculi, Ⅲ: Semantical campleteness of some many-valued prepositional calculi, Zeitschr, F. Math. Logic and Grundlagend. Math., 25(1979)45-52, 119-134,447-353.
[21] 邹开其，徐扬，模糊系统与专家系统，西南交通大学出版社．1989．
[22] 徐扬，格蕴涵代数，西南交通大学学报，1(1993)，20-27．
[23] 徐扬，格蕴涵代数中的同态，中国第五届多值逻辑学术会议论文集，(1992)206-210．
[24] 黄天民，徐扬，赵海良，吴建乐，格、序引论及其应用，西南交通大学出版社，(1998)135-150．
[25] 宋振明，格H蕴涵代数的一个表示，兰州大学学报，32(1996)349-351．
[26] Song Zhenming, Prime space of lattice implication algebras, Chinese Quarterly Journal of Mathematics, 12(1997)46-50.
[27] 刘军，关于格蕴涵代数结构的研究，西南交通大学硕士论文，1996．
[28] 王学芳，关于格蕴涵代数性质和结构的研究，西南交通大学硕士论文，2001．
[29] XuYang and Qin Keryun,Lattice properties of lattice implication algebra,Symposium on Applied Mathematics,Chengdu Science and Technology University Press,(1992)54-58.
[30] Xu Yang and Qin Keyun, On filter of lattice implication algebras, The Journal of Fuzzy Mathematics,2(1993)251-260.
[31] Young Bae Jun, Eun Hwan Roh and Yang Xu, LI-ideals in lattice
implication algebras, Bull Korean Math. Soc. 35(1998), No.1.pp.13-24.
[32]宋振明，徐扬，格蕴涵代数上的同余关系，应用数学，10(1997)，121-124．
[33]马骏，关于格蕴涵代数和格值命题逻辑系统LP(X)的研究，西南交通大学硕士论文，1999．
[34]徐扬，秦克云，格H蕴涵代数与格蕴涵代数类，河北煤炭建筑工程学院学报，3(1992)，139-143．
[2] 宋振明，基于格蕴涵代数的智能控制中不确定性推理的研究，西南交通大学博士论文，(1998)14-15．
[3] J.Lukasiewicz, On 3-valued logic, Ruch Filozoficzny, 5(1920)169-170.
[4] 秦克云，基于格蕴涵代数的格值逻辑系统及应用的研究，西南交通大学博士论文，(1996)2-6．
[5] D.A.Bochvav, On a 3-valued logic calculus and its application to the analysis of contradictions matematiceskij sbornik, 4(1939),287-308.
[6] 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，西南交通大学博士论文，(1998)3-8．
[7] E.Post,Introduction to a general theory of elementary propositions,American Journal of Mathematics, 43(1921)163-185.Reprinted in J.Van Heijenoort(ed.),From Frege to Godel:A source book in Mathematical Logic 1879-1931 ,Cambridge,Mass, 1967.
[8] Leonard Bolc, Piotr Borowik, Many-valued Logics, Springer-Verlag, Berlin Heidelberg New York, 1992.
[9] L.A.Zadeh, Fuzzy sets, Inf. Control, 8(1965),338-353.
[10] G.Birkhoff, Lattice Theory, 3rd edition, American Mathematical Society, Proridence, R.L,1967.
[11] J.A.Goguen,The logic of inexact concepts, Synthese, 19(1969),325-373.
[12] J.Lukasiewicz, Selected Works-Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam/PWN,Warzawa, 1971.
[13] Saunders Mac Lane, Categories for the Working Mathematician, Springer Verlag Berlin, 1971.
[14] C.C.Chang, Keisler H.J.,Model Theory, North-Holland Publishing Company, Amsterdam, 1973.
[15] Zhang Jinwen,A unified treatment of fuzzy set theory and boolean valued set theory-fuzzy set structures and nomal fuzzy set structure, Journal of Mathematical Analysis and Applications,76(1980),297-301.
[16] C.C.Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc. 88(1958)195-234.
[17] P. Mangani,On certain algebras related to many-valued logics, Boll.V.M.I 8(4)(1973)68-78.
[18] X.H.Liu, H.Xiao, Operator fuzzy logic and fuzzy resolution, Proc. Of the 5th IEEE Inter. Symp. on Multiple-valued Logic, ISMUL 85, Kingstion, Canada, (1985)68-75.
[19] K.Lano,Fuzzy sets and residuated logic, Fuzzy Sets and System 47(2)(1969)203-220.
[20] J.Pavelka, On fuzzy logic Ⅰ:Many-valued rules of inference, Ⅱ: Enriched residuated lattices and semantics of prepositional calculi, Ⅲ: Semantical campleteness of some many-valued prepositional calculi, Zeitschr, F. Math. Logic and Grundlagend. Math., 25(1979)45-52, 119-134,447-353.
[21] 邹开其，徐扬，模糊系统与专家系统，西南交通大学出版社．1989．
[22] 徐扬，格蕴涵代数，西南交通大学学报，1(1993)，20-27．
[23] 徐扬，格蕴涵代数中的同态，中国第五届多值逻辑学术会议论文集，(1992)206-210．
[24] 黄天民，徐扬，赵海良，吴建乐，格、序引论及其应用，西南交通大学出版社，(1998)135-150．
[25] 宋振明，格H蕴涵代数的一个表示，兰州大学学报，32(1996)349-351．
[26] Song Zhenming, Prime space of lattice implication algebras, Chinese Quarterly Journal of Mathematics, 12(1997)46-50.
[27] 刘军，关于格蕴涵代数结构的研究，西南交通大学硕士论文，1996．
[28] 王学芳，关于格蕴涵代数性质和结构的研究，西南交通大学硕士论文，2001．
[29] XuYang and Qin Keryun,Lattice properties of lattice implication algebra,Symposium on Applied Mathematics,Chengdu Science and Technology University Press,(1992)54-58.
[30] Xu Yang and Qin Keyun, On filter of lattice implication algebras, The Journal of Fuzzy Mathematics,2(1993)251-260.
[31] Young Bae Jun, Eun Hwan Roh and Yang Xu, LI-ideals in lattice
implication algebras, Bull Korean Math. Soc. 35(1998), No.1.pp.13-24.
[32]宋振明，徐扬，格蕴涵代数上的同余关系，应用数学，10(1997)，121-124．
[33]马骏，关于格蕴涵代数和格值命题逻辑系统LP(X)的研究，西南交通大学硕士论文，1999．
[34]徐扬，秦克云，格H蕴涵代数与格蕴涵代数类，河北煤炭建筑工程学院学报，3(1992)，139-143．