Script Z sign-Continuous Posets and Their Topological Manifestation

Applied Categorical Structures

Volumn 7, Issue 1-2, 1999, Pages 31-70

  Erné, Marcel ^a  

^a LEIBNIZ UNIVERSITY HANNOVER   (Germany)

Author keywords

Closure space; Compact; Completely distributive; Completion; Continuous poset; Core; Sober space

Indexed keywords

EID: 0007468993     PISSN: 09272852     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1008657800278     Document Type: Article

References (56)

1. Adámek, J.: Construction of free continuous algebras, Algebra Universalis 14 (1982), 140-166.
2. Adámek, J., Herrlich, H. and Strecker, G.: Abstract and Concrete Categories, Wiley Interscience, New York, 1990.
3. Alexandroff, P.: Diskrete Räume, Mat. Sb. (N.S.) 2 (1937), 501-518.
4. Banaschewski, B.: Hüllensysteme und Erweiterungen von Quasi-Ordnungen, Z. Math. Logik Grundlagen Math. 2 (1956), 369-377.
5. Banaschewski, B. and Nelson, E.: Completions of partially ordered sets, SIAM J. Comput. 11 (1982), 521-528.
6. Banaschewski, B. and Hoffmann, R.-E. (eds.): Continuous Lattices. Bremen 1979, Lecture Notes in Math. 871, Springer-Verlag, Berlin, Heidelberg, New York, 1981.
7. Bandelt, H.-J. and Erné, M.: The category of Z-continuous posets, J. Pure Appl. Algebra 30 (1983), 219-226.
8. Bandelt, H.-J. and Erné, M.: Representations and embeddings of M-distributive lattices, Houston J. Math. 10 (1984), 315-324.
9. Birkhoff, G.: Rings of sets, Duke Math. J. 3 (1937), 443-454.
10. Bruns, G.: Darstellungen und Erweiterungen geordneter Mengen I, II, J. Reine Angew. Math. 209 (1962), 167-200; 210 (1962), 1-23.
11. Bruns, G.: Darstellungen und Erweiterungen geordneter Mengen I, II, J. Reine Angew. Math. 209 (1962), 167-200; 210 (1962), 1-23.
12. Dobbertin, H., Erné, M. and Kent, D. C.: A note on order convergence in complete lattices, Rocky Mountain J. Math. 14 (1984), 647-654.
13. Erné, M.: A completion-invariant concept of continuous lattices, in: [6], pp. 34-60.
14. Erné, M.: Scott convergence and Scott topologies on partially ordered sets II, in: [6], pp. 61-96.
15. Erné, M.: Homomorphisms of M-distributive and M-generated posets, Tech. Report No. 125, Institut für Mathematik, Universität Hannover, 1981.
16. Erné, M.: Adjunctions and standard constructions for partially ordered sets, in: G. Eigenthaler et al. (eds.), Contributions to General Algebra 2. Proc. Klagenfurt Conf. 1982, Hölder-Pichler-Tempsky, Wien, 1983, pp. 77-106.
17. Erné, M.: Chains, directed sets and continuity, Tech. Report No. 175, Institut für Mathematik, Universität Hannover, 1984.
18. Erné, M.: Order extensions as adjoint functors, Quaestiones Math. 9 (1986), 146-204.
19. Erné, M.: The ABC of order and topology, in: H. Herrlich and H.-E. Porst (eds.), Category Theory at Work, Heldermann, Berlin, 1991, pp. 57-83.
20. Erné, M.: The Dedekind-MacNeille completion as a reflector, Order 8 (1991), 159-173.
21. Erné, M.: Bigeneration in complete lattices and principal separation in partially ordered sets, Order 8 (1991), 197-221.
22. Erné, M.: Algebraic ordered sets and their generalizations, in: I. Rosenberg and G. Sabidussi (eds.), Algebras and Orders. Proc. Montreal 1992, Kluwer, Amsterdam, 1994.
23. Erné, M., Koslowski, J., Melton, A. and Strecker, G.: A primer on Galois connections, in: S. Andima et al. (eds.), Papers on General Topology and Its Applications. 7th Summer Conf. Wisconsin, Annals New York Acad. Sci. 704, New York, 1993, pp. 103-125.
24. Erné, M. and Wilke, G.: Standard completions for quasiordered sets, Semigroup Forum 27 (1983), 351-376.
25. Ershov, Y. L.: The theory of A-spaces, Algebra and Logic 12 (1974), 209-232. Translated from: Algebra i Logika 12 (1973), 369-416.
26. Ershov, Y. L.: The theory of A-spaces, Algebra and Logic 12 (1974), 209-232. Translated from: Algebra i Logika 12 (1973), 369-416.
27. Fleischer, I.: Even every join-extension solves a universal problem, J. Austral. Math. Soc. 21 (1976), 220-223.
28. Frink, O.: Ideals in partially ordered sets, Amer. Math. Monthly 61 (1954), 223-234.
29. Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S.: A Compendium of Continuous Lattices, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
30. Hoffmann, R.-E.: Projective sober spaces, in: [6], pp. 125-158.
31. Hoffmann, R.-E.: Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications, in: [6], pp. 159-208.
32. Hoffmann, R.-E. and Hofmann, K. H. (eds.): Continuous Lattices and Their Applications, Bremen 1982, Lecture Notes in Pure and Appl. Math. 101, Marcel Dekker, New York, 1985.
33. Hofmann, K. H. and Mislove, M.: Free objects in the category of completely distributive lattices, in: [30], pp. 129-151.
34. Isbell, J. R.: Completion of a construction of Johnstone, Proc. Amer. Math. Soc. 85 (1982), 333-334.
35. Iwamura, T.: A lemma on directed sets, Zenkoku Shijo Sugaku Danwakai 262 (1944), 107-111 (Japanese).
36. Johnstone, P. T.: Stone Spaces, Cambridge University Press, Cambridge, 1982.
37. Johnstone, P. T.: Scott is not always sober, in: [6], pp. 282-283.
38. Kříž, I. and Pultr, A.: A spatiality criterion and a quasitopology which is not a topology, Houston J. Math. 15 (1989), 215-233.
39. Lawson, J. D.: The duality of continuous posets, Houston J. Math. 5 (1979), 357-386.
40. Lawson, J. D.: The versatile continuous order, in: Lecture Notes in Comp. Sci. 298, Springer-Verlag, Berlin, Heidelberg, New York, 1987, pp. 134-160.
41. Markowsky, G.: A motivation and generalization of Scott's notion of a continuous lattice, in: [6], pp. 298-307.
42. Markowsky, G.: Propaedeutic to chain-complete posets with basis, in: [6], pp. 308-314.
43. Meseguer, J.: Order completion monads, Algebra Universalis 16 (1983), 63-82.
44. Nelson, E.: Z-continuous algebras, in: [6], pp. 315-334.
45. Novak, D.: Generalization of continuous posets, Trans. Amer. Math. Soc. 272 (1982), 645-667.
46. Novak, D.: On a duality between the concepts 'finite' and 'directed', Houston J. Math. 8 (1982), 545-563.
47. Papert, S.: Which distributive lattices are lattices of closed sets? Proc. Cambridge Philos. Soc. 55 (1959), 172-176.
48. Raney, G. N.: A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518-522.
49. Schmidt, J.: Each join-completion of a partially ordered set is the solution of a universal problem, J. Austral. Math. Soc. 17 (1974), 406-413.
50. Scott, D. S.: Continuous lattices, in: Toposes, Algebraic Geometry and Logic, Lecture Notes in Math. 274, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
51. Scott, D. S.: Domains for denotational semantics, in: H. Nielsen and E. M. Schmidt (eds.), ICALP 82, Lecture Notes in Comp. Sci. 140, Springer-Verlag, Berlin, Heidelberg, New York, 1982.
52. Shi, G. B.: A new characterization of Z-continuous posets, Preprint Louisiana State University, Baton Rouge, 1996.
53. Venugopalan, G.: Z-continuous posets, Houston J. Math. 12 (1986), 275-294.
54. Wright, J. B., Wagner, E. G. and Thatcher, J. W.: A uniform approach to inductive posets and inductive closure, Theor. Comp. Sci. 1 (1978), 57-77.
55. Wyler, O.: Dedekind complete posets and Scott topologies, in: [6], pp. 384-389.
56. Xu, X.-Q.: Embeddings of strongly Z-precontinuous posets in cubes, Submitted to J. Pure and Appl. Algebra.