## Abstract and Applied Analysis

### Some New Intrinsic Topologies on Complete Lattices and the Cartesian Closedness of the Category of Strongly Continuous Lattices

#### Abstract

We prove some new characterizations of strongly continuous lattices using two new intrinsic topologies and a class of convergences. Lastly we show that the category of strongly continuous lattices and Scott continuous mappings is cartesian closed.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 942628, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511769

Digital Object Identifier
doi:10.1155/2013/942628

Mathematical Reviews number (MathSciNet)
MR3035270

Zentralblatt MATH identifier
1288.06014

#### Citation

Wu, Xiuhua; Li, Qingguo; Zhao, Dongsheng. Some New Intrinsic Topologies on Complete Lattices and the Cartesian Closedness of the Category of Strongly Continuous Lattices. Abstr. Appl. Anal. 2013 (2013), Article ID 942628, 8 pages. doi:10.1155/2013/942628. https://projecteuclid.org/euclid.aaa/1393511769

#### References

• D. Scott, “Continuous Lattices,” in Toposes, Algebraic Geometry and Logic (Conf., Dalhousie Univ., Halifax, N. S., 1971), vol. 274 of Lecture Notes in Math., pp. 97–136, Springer, Berlin, Germany, 1972.
• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains, Springer, Berlin, Germany, 2003.
• A. Jung, “Cartesian closed categories of algebraic cpos,” Theoretical Computer Science, vol. 70, no. 2, pp. 233–250, 1990.
• G. Gierz and J. D. Lawson, “Generalized continuous and hypercontinuous lattices,” The Rocky Mountain Journal of Mathematics, vol. 11, no. 2, pp. 271–296, 1981.
• D. Zhao, “Semicontinuous lattices,” Algebra Universalis, vol. 37, no. 4, pp. 458–476, 1997.
• R. C. Powers and T. Riedel, “$Z$-semicontinuous posets,” Order, vol. 20, no. 4, pp. 365–371, 2003.
• Y. Liu and L. Xie, “On the category of semicontinuous lattices,” Journal of Liaoning Normal University (Natural Science), vol. 20, no. 3, pp. 182–185, 1997 (Chinese).
• Y. Liu and L. Xie, “On the structure of semicontinuous lattices and cartesian closedness of categories of semicontinuous lattices,” Journal of Liaoning Normal University (Natural Science), vol. 18, no. 4, pp. 265–268, 1995 (Chinese).
• X. H. Wu, Q. G. Li, and R. F. Xu, “Some properties of semicontinuous lattices,” Fuzzy Systems and Mathematics, vol. 20, no. 4, pp. 42–46, 2006 (Chinese).
• X. H. Wu and Q. G. Li, “Characterizations and functions of semicontinuous lattices,” Journal of Mathematical Research and Exposition, vol. 27, no. 3, pp. 654–658, 2007 (Chinese).
• Y. Rav, “Semiprime ideals in general lattices,” Journal of Pure and Applied Algebra, vol. 56, no. 2, pp. 105–118, 1989.
• G. N. Raney, “Completely distributive complete lattices,” Proceedings of the American Mathematical Society, vol. 3, pp. 677–680, 1952.
• B. Zhao and D. Zhao, “Lim-inf convergence in partially ordered sets,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 701–708, 2005.
• M. B. Smyth, “The largest Cartesian closed category of domains,” Theoretical Computer Science, vol. 27, no. 1-2, pp. 109–119, 1983.