M-SCOTT CONVERGENCE AND M-SCOTT TOPOLOGY ON POSETS

• Journal title : Honam Mathematical Journal
• Volume 33, Issue 2,  2011, pp.279-300
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2011.33.2.279
Title & Authors
M-SCOTT CONVERGENCE AND M-SCOTT TOPOLOGY ON POSETS
Yao, Wei;

Abstract
For a subset system M on any poset, M-Scott notions, such as M-way below relation,M-continuity,M-Scott convergence (of nets and filters respectively) and M-Scott topology are proposed Any approximating auxiliary relation on a poset can be represented by an M-way below relation such that this poset is M-continuous. It is shown that a poset is M-continuous iff the M-Scott topology is completely distributive. The topology induced by the M-Scott convergence coincides with the M-Scott topology. If the M-way below relation satisfies the property of interpolation then a poset is M-continuous if and only if the M-Scott convergence coincides with the M-Scott topological convergence. Also, M-continuity is characterized by a certain Galois connection.
Keywords
M-way below;M-continuous;M-Scott topology;M-Scott convergence;
Language
English
Cited by
References
1.
B. Banaschewski, R.-E. Hoffmann (Eds.), Continuous Lattices, Bremen 1979, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin/Heidelberg/New York, 1981.

2.
H.-J. Bandelt, M-distributive lattices, Arch. Math. 39 (1982) 436-442.

3.
H.-J. Bandelt, M. Erne, The category of Z-continuous posets, J. Pure Appl. Algebra 30 (1983) 219-226.

4.
H.-J. Bandelt, M. Erne, Representations and embedding of M-distributive lattices, Houston J. Math. 10 (1984) 315-324.

5.
A. Baranga, Z-continuous poset, Discrete Mathematics 152 (1996) 33-45.

6.
M. Erne, Completion-invariant extension of the concept of continuous lattice, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin/Heidelberg/New York, 1981, pp. 45-60.

7.
M. Erne, Homomorphisms of M-distributive and M-generated posets, Tech. Report No. 125, Institut fur Mathematik, Universitat, Hannover, 1981, pp. 315-324.

8.
M. Erne, Scott convergence and Scott topology on partially ordered sets II, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin/Heidelberg/New York, 1981, pp. 61-96.

9.
M. Erne, Z-continuous posets and their topological manifestation, Applied Categorical Structures 7 (1999) 31-70.

10.
O. Frink, Ideals in partially ordered sets, Amer. Math. Monthly 61 (1954) 223-234.

11.
G. Gierz, et al, A Compendium of Continuous Lattices, Springer-Verlag, New York, 1980.

12.
G. Gierz, et al, Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.

13.
G. Gratzer, General Lattice Theory (2nd edition), Birkhauser, Basel-Boston-Berlin, 1998.

14.
R.-E. Hoffmann, Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin/Heidelberg/New York, 1981, pp. 159-108.

15.
R.-E. Hoffmann, K.H. Hofmann (Eds.), Continuous Lattices and Their Applications, Bremen 1982, Lecture Notes in Pure and Applied Mathematics 101, Marcel Dekker, New York, 1985.

16.
D.C. Kent, Convergence functions and their related topologies, Fund. Math. 54 (1964) 125-133.

17.
J.D. Lawson, The duality of continuous posets, Houston J. Math. 5 (1979) 357-386.

18.
J.H Liang, K. Keimel, Order enviroments of topological spaces, Acta Mathematica Sinica 20 (2004) 943-948.

19.
E. Nelson, Z-continuous poset, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin/Heidelberg/New York, 1981, pp. 315-334.

20.
D. Novak, Generalization of continuous posets, Trans. Amer. Math. Soc. 272 (1982) 645-667.

21.
O. Ore, Galois connexions, Trans. Amer. Math. Soc. 55 (1944) 493-513.

22.
D.S. Scott, Outline of a mathematical theory of computation, Proc. 4th Annual Princeton Conf. on Information Science and Systems, Princeton University Press, Princeton, NJ, 1970, pp. 169-176.

23.
D.S. Scott, Continuous lattices, Topos, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer-Verlag, Berlin, 1972, pp. 97-136.

24.
G.B. Shi, A new characterization of Z-continuous posets, Preprint Louisiana State University, Baton Rouge, 1996.

25.
J. Schmidt, Beitrage zur filter theorie II, Math. Nachr. 10 (1953) 199-232.

26.
P. Venugopalan, Z-continuous posets, Houston J. Math. 12 (1986) 275-294.

27.
P. Venugopalan, A generalization of completely disdributive lattices, Algebra Universalis 27 (1990) 578-586.

28.
G.-J. Wang, Theory of $\phi$-minimal sets and its applications, Chinese Sci. Bull. 32 (1987) 511-516.

29.
G.-J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems 47 (1992) 351-376.

30.
S. Weck, Scott convergence and Scott topology in partially ordered sets I, Lecture Notes in Mathematics 871, Springer-Verlag, Berlin/Heidelberg/New York, 1981, pp. 372-383.

31.
J.B. Wright, E.G. Wagner, J.W. Thatcher, A uniform approach to inductive posets and inductive closure, Lecture Notes in Comupter Science 53, Springer-Verlag, Berlin-New York, 1977, pp. 192-212.

32.
J.B. Wright, E.G. Wagner, J.W. Thatcher, A uniform approach to inductive posets and inductive closure, Theoretical Computer Science 7 (1978) 57-77.

33.
L.S. Xu, Continuity of posets via Scott topology and sobrification, Topology and its Applications 153 (2006) 1886-1894.

34.
X.-Q. Xu, Constructions of homomorphisms of M-continuous lattices, Trans. Amer. Math. Soc. 347 (1995) 3167-3175.

35.
W. Yao, L.-X. Lu, Relationships between Galois connections and operators, The 4th International Symposium on Domain Theory, 2006, June 1-5, Changsha, China, pp. 43-46.

36.
B. Zhao, D.S. Zhao, Liminf convergence in partially ordered posets, J. Math. Anal. Appl. 309 (2005) 701-708.

37.
Y. Zhou, B. Zhao, Order-convergence and $lim-inf_M$-convergence in posets, J. Math. Anal. Appl. 325 (2007) 655-664.