超格上的商结构理论
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摘要
超代数系统理论是非常重要的现代数学分支，许多专家学者对其进行了深入细致的系统研究。本文主要在超格上引入了加法(乘法)扩张、对换超格等概念，在此基础上构建出商超格，研究商超格的一些性质，更进一步地建立了第一、第二、第三同构定理。同时也对格蕴涵代数上的强LI-理想做了初步的讨论。
第一章，我们对超格理论研究的背景、现状和格的基础知识作了简要的介绍。第二、三章，我们分别研究超格上的商代数和超格上的商结构，以及相关的一些性质。利用引入超格的乘法(加法)扩张，对换超格等概念，引出超格的陪集，构造出商超格。在此基础上，也进一步研究了商超格的对换性和商超格的其它性质。第四章，研究了格蕴涵代数的强LI-理想，并讨论了它的一些性质。
The theory of hyperalgebra is a very important class of mathematics. Many experts and scholars thorough, painstakingly and systematically investigate it. In this paper we mainly introduce the concepts of multiplication (or addition) extension and transposition. The quotient hyperlattices is constructed, and some properties of quotient hyperlattices is also studied. Moreover the First, Second and Third Isomorphism Theorems have been established.
In chapter 1, we simply introduce the background and the present state of the theory of the hyperlattice; for the sake of convenience, we introduce some fundamental knowledge about lattice. In chapter 2, 3 we study the quotient algebras of hyperlattice and the quotient structures of hyperlattice , and some properties of quotient hyperlattices. According to introducing the concepts of multiplication (or addition) extension and transposition, the cosets of hyperlattice is studied and the quotient hyperlattices is constructed. On the basis, we study the transposition of quotient hyperlattices and other properties of quotient hyperlattices. In chapter 4 we study the strong Li-ideals in lattice implication algebra. Some properties of the strong Li-ideals in lattice implication algebra is discussed.
第一章，我们对超格理论研究的背景、现状和格的基础知识作了简要的介绍。第二、三章，我们分别研究超格上的商代数和超格上的商结构，以及相关的一些性质。利用引入超格的乘法(加法)扩张，对换超格等概念，引出超格的陪集，构造出商超格。在此基础上，也进一步研究了商超格的对换性和商超格的其它性质。第四章，研究了格蕴涵代数的强LI-理想，并讨论了它的一些性质。
The theory of hyperalgebra is a very important class of mathematics. Many experts and scholars thorough, painstakingly and systematically investigate it. In this paper we mainly introduce the concepts of multiplication (or addition) extension and transposition. The quotient hyperlattices is constructed, and some properties of quotient hyperlattices is also studied. Moreover the First, Second and Third Isomorphism Theorems have been established.
In chapter 1, we simply introduce the background and the present state of the theory of the hyperlattice; for the sake of convenience, we introduce some fundamental knowledge about lattice. In chapter 2, 3 we study the quotient algebras of hyperlattice and the quotient structures of hyperlattice , and some properties of quotient hyperlattices. According to introducing the concepts of multiplication (or addition) extension and transposition, the cosets of hyperlattice is studied and the quotient hyperlattices is constructed. On the basis, we study the transposition of quotient hyperlattices and other properties of quotient hyperlattices. In chapter 4 we study the strong Li-ideals in lattice implication algebra. Some properties of the strong Li-ideals in lattice implication algebra is discussed.
引文
[1] 阮传概．近世代数及其应用[M]．北京：邮电大学出版社，1998．
[2] 黄天发．离散数学[M]．四川：电子科技大学出版社，1995．
[3] 傅彦，顾小丰．离散数学及其应用[M]．北京：电子工业出版社，1997．
[4] 范云．近世代数[M]．北京：高等教育出版社，1988．
[5] 吴品三．近世代数[M]_北京：人民教育出版社，1983．
[6] 李盘林，李丽双，李洋，王春丽．离散数学[M]．北京：高等教育出版社，1999．
[7] 盛德成．抽象代数[M]．北京：科学出版社，2000．
[8] 莫宗坚等．代数学(上)[M]．北京：北京大学出版社，1987．
[9] 赵彬．分配超格[J]．西北大学学报(自然科学版)，2005，35(2)：125-129．
[10] 郭效芝，辛小龙．超格[J]．纯粹数学与应用数学，2004，20(1)：40-43．
[11] 郑崇友，樊磊，崔宏斌．Frame与连续格[M．北京：首都师范大学出版社，2000．
[12] 中山正(董克诚译)．格论[M]．上海科技出版社，1964．
[13] 陈杰．格论初步[M]．内蒙古大学出版社，1990．
[14] 黄崇智．关于同态及同构[J]．内江师范学院学报，2002，17(6)：62-66．
[15] 刘绍学，朱元森．数学辞海．山西教育出版社，2002，8．
[16] L.L.Yong,Y.L.San.ILI-ideals and prime LI-ideals in lattice implication algebras[J]. Information Sciences, 2003,155:157-175.
[17] G. Birkhoff. Lattice Theory, American Mathematical Society Colloquium, 1940.
[18] B.J.Yong, X.L.Xin.Strong Hyperbck-ideals of Hyperbck-Algebras[J].Math. Japonica 51,No. 3(2000),493-498
[19] J.Meng,X.L.Xin.Quotient BCK-algebra Induced by a Fuzzy Ideal[J]. Southeast Asian Bulletin of Mathematics(1999)23:243-251
[20] Rosaria Rota. Hyperaffine planes over hypcrrings [J]. Discrete Mathematices,1996,155(3): 215-223.
[21] H. s. Wall. Hypergroups. AMer. J. Math. 1973, 59: 77-98.
[22] James Jantosciak. Transposition Hypergroups. Noncommutative Join Spaces[J]. Journal of algebra, 1997, 187(1):97-119.
[23] Y. B. Jun, M. M. Zahedi, X. L. Xin, R.A. Borzoei. On hyperBCK-algebras. Italian[J]. Pure and Appl. Math. ,2000, 8: 127-136.
[24] M. I. Kargapolov, J. I. Merzliakov. Fundamentals of the theory of groups. SpringerVerlag, New York, 1979.
[25] T. M. Apostal. Introduction to Analytic Number Theory[M]. New York: SpringerVerlag, 1976.
[26] T. w. Hungerford. Algebras. Springer-Verlag, New York(1974).
[27] Marty F., Surne generalization de la notion de group [A], 8congress Math[C], Scandinaves, Stockholm, 1934, 45-49.
[28] Cui H B,Zheng C Y. Stratification Structures on a Kind of Completely Distributive Lattices and Their Applications in the Theory of Topological Molecular Lattices[J]. Fuzzy Sets and Syetems, 1999, 106: 449-454.
[29] Xin X L. On fuzzy hypersubalgebras of hyper BCK-algebras[J]. Pure and Applied Mathematics[J]. 2001, 17(3): 261-266.
[30] Corsini P. Prolegomena of hypergroup theory[M]. Aviani Editore, 1993.
[31] Jun Y B,Xin X L. Scalar elements and hyperatoms of hyper BCK-algebras[J]. Scientiae Math., 1999,2: 303-309.
[32] Jun Y B, Xin X L. Positive implicative hyper BCK-algebras[J]. Czech. Math. J. (submitted)
[33] K. Takeuchi. On maximal proper sublattices. Journal Math. Soc. Japan, 1951, 51: 348-359.
[34] G. Gratzer, E. T. Schmidt. Two notes on lattice congruences. Annales Univ. Sci. Budapestiensis de Rolando Eotvos nominatae, Sectio math., 19568,1:206-216.
[35] Xiao Long Xin. hyper BCI-algeras. Disussiones Mathematicate General Algebra ang Applications, 2006, 26: 5-9.
[36] Rata R.Sugli iperanelli maltiplicativi [J]. Rend Mathematics Roma, 1982, (4):36-42.
[37] Procesi R, Rota R. On some classes of hyperstructures [J]. Discrete Mathematics, 1999, 208/209:485-497.
[38] X.L.Xin, Hyper BCI-algebras, DiscussionES Mathematicas General Algebra and. Applications 26(2006),5-19.
[2] 黄天发．离散数学[M]．四川：电子科技大学出版社，1995．
[3] 傅彦，顾小丰．离散数学及其应用[M]．北京：电子工业出版社，1997．
[4] 范云．近世代数[M]．北京：高等教育出版社，1988．
[5] 吴品三．近世代数[M]_北京：人民教育出版社，1983．
[6] 李盘林，李丽双，李洋，王春丽．离散数学[M]．北京：高等教育出版社，1999．
[7] 盛德成．抽象代数[M]．北京：科学出版社，2000．
[8] 莫宗坚等．代数学(上)[M]．北京：北京大学出版社，1987．
[9] 赵彬．分配超格[J]．西北大学学报(自然科学版)，2005，35(2)：125-129．
[10] 郭效芝，辛小龙．超格[J]．纯粹数学与应用数学，2004，20(1)：40-43．
[11] 郑崇友，樊磊，崔宏斌．Frame与连续格[M．北京：首都师范大学出版社，2000．
[12] 中山正(董克诚译)．格论[M]．上海科技出版社，1964．
[13] 陈杰．格论初步[M]．内蒙古大学出版社，1990．
[14] 黄崇智．关于同态及同构[J]．内江师范学院学报，2002，17(6)：62-66．
[15] 刘绍学，朱元森．数学辞海．山西教育出版社，2002，8．
[16] L.L.Yong,Y.L.San.ILI-ideals and prime LI-ideals in lattice implication algebras[J]. Information Sciences, 2003,155:157-175.
[17] G. Birkhoff. Lattice Theory, American Mathematical Society Colloquium, 1940.
[18] B.J.Yong, X.L.Xin.Strong Hyperbck-ideals of Hyperbck-Algebras[J].Math. Japonica 51,No. 3(2000),493-498
[19] J.Meng,X.L.Xin.Quotient BCK-algebra Induced by a Fuzzy Ideal[J]. Southeast Asian Bulletin of Mathematics(1999)23:243-251
[20] Rosaria Rota. Hyperaffine planes over hypcrrings [J]. Discrete Mathematices,1996,155(3): 215-223.
[21] H. s. Wall. Hypergroups. AMer. J. Math. 1973, 59: 77-98.
[22] James Jantosciak. Transposition Hypergroups. Noncommutative Join Spaces[J]. Journal of algebra, 1997, 187(1):97-119.
[23] Y. B. Jun, M. M. Zahedi, X. L. Xin, R.A. Borzoei. On hyperBCK-algebras. Italian[J]. Pure and Appl. Math. ,2000, 8: 127-136.
[24] M. I. Kargapolov, J. I. Merzliakov. Fundamentals of the theory of groups. SpringerVerlag, New York, 1979.
[25] T. M. Apostal. Introduction to Analytic Number Theory[M]. New York: SpringerVerlag, 1976.
[26] T. w. Hungerford. Algebras. Springer-Verlag, New York(1974).
[27] Marty F., Surne generalization de la notion de group [A], 8congress Math[C], Scandinaves, Stockholm, 1934, 45-49.
[28] Cui H B,Zheng C Y. Stratification Structures on a Kind of Completely Distributive Lattices and Their Applications in the Theory of Topological Molecular Lattices[J]. Fuzzy Sets and Syetems, 1999, 106: 449-454.
[29] Xin X L. On fuzzy hypersubalgebras of hyper BCK-algebras[J]. Pure and Applied Mathematics[J]. 2001, 17(3): 261-266.
[30] Corsini P. Prolegomena of hypergroup theory[M]. Aviani Editore, 1993.
[31] Jun Y B,Xin X L. Scalar elements and hyperatoms of hyper BCK-algebras[J]. Scientiae Math., 1999,2: 303-309.
[32] Jun Y B, Xin X L. Positive implicative hyper BCK-algebras[J]. Czech. Math. J. (submitted)
[33] K. Takeuchi. On maximal proper sublattices. Journal Math. Soc. Japan, 1951, 51: 348-359.
[34] G. Gratzer, E. T. Schmidt. Two notes on lattice congruences. Annales Univ. Sci. Budapestiensis de Rolando Eotvos nominatae, Sectio math., 19568,1:206-216.
[35] Xiao Long Xin. hyper BCI-algeras. Disussiones Mathematicate General Algebra ang Applications, 2006, 26: 5-9.
[36] Rata R.Sugli iperanelli maltiplicativi [J]. Rend Mathematics Roma, 1982, (4):36-42.
[37] Procesi R, Rota R. On some classes of hyperstructures [J]. Discrete Mathematics, 1999, 208/209:485-497.
[38] X.L.Xin, Hyper BCI-algebras, DiscussionES Mathematicas General Algebra and. Applications 26(2006),5-19.