|
|
- 2016
Topos中偏序对象的上(下)确界
|
Abstract:
摘要： 基于topos中的偏序对象, 讨论了topos中的上下确界的定义, 推广了格论中的经典结论, 将一些格论中熟知的结论提升到了topos中。 得到了主要结果:topos中的对象为完备偏序对象当且仅当该对象的任意广义子集存在上(下)确界。
Abstract: The concepts of supremum and infimum which are common generalizations of lattice theory is introduced, and some well-known lattice theory into an arbitrary topos is lifted. The main result that an object in a topos is a complete partially ordered object if and only if its general subset has supremum and infimum is obtianed
| [1] | 孟晓青. Topos理论简介[J]. 数学进展, 1992, 21(1):1-24. MENG Xiaoqing. An introduction to topos theory[J]. Advances in Mathematics, 1992, 21(1):1-24. |
| [2] | LUO Maokang, HE Wei. A new logic for uncertainty[J]. 2015, http://arxiv.org/abs/1506.03123. |
| [3] | 卢涛,王习娟,贺伟.Topos中的完备偏序对象上的算子理论[J].山东大学学报(理学版),2016,51(2):64-71. LU Tao, WANG Xijuan, HE Wei. The operator theory on complete partially ordered objects in a topos[J]. Journal of Shandong University(Natural Science), 2016, 51(2):64-71. |
| [4] | JOHNSTONE P T, JOYAL A. Continuous categories and exponentiable toposes[J]. Journal of Pure and Applied Algebra, 1982, 25:255-296. |
| [5] | KOCK A, LECOUTURIER P, MIKKELSEN C J. Some topos theoretic concepts of finiteness[J]. Lecture Notes in Math. Berlin: Springer-Verlag, 1975, 445:209-283. |
| [6] | HE Wei, LUO Maokang. Quantum spaces[J]. Acta Mathematica Sinica, English Series, 2010, 26(7):1323-1330. |
| [7] | MAC LANE S, MOERDIJK L. Sheaves in geometry and logic: a first introduction to topos[M]. New York: Springer-Verlag, 1992. |
| [8] | MAC LANE S. Categories for working mathematician[M]. New York: Springer-Verlag, 1972. |
| [9] | 卢涛, 贺伟, 王习娟. Galois connections in a topos[J]. 数学研究与评论, 2010, 30(3):381-389. LU Tao, HE Wei, WANG Xi Juan. Galois connections in a topos[J]. Journal of Mathematical Research and Exposition, 2010, 30(3):381-389. |
| [10] | 王习娟, 贺伟. On the heyting algebra objects in a topos[J]. Journal of Mathematics, 2011, 31(6):979-998. WANG Xijuan, HE Wei. On the heyting algebra objects in a topos[J]. 数学杂志, 2011, 31(6):979-998. |
| [11] | 贺伟. 范畴论[M]. 北京: 科学出版社, 2006. HE Wei. Category theory[M]. Beijing: Science Press, 2006. |
| [12] | JOHNSTONE P T. Sketches of an elephant: a topos theory compendium[M]. Oxford: Oxford University Press, 2002. |
