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The Injective Hull and the -Compactification of a Continuous Poset

Published online by Cambridge University Press:  20 November 2018

Rudolf-E. Hoffmann
Rudolf-E. Hoffmann*
Affiliation:
Universität Bremen, Bremen, Federal Republic of Germany
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In [57] (2.12), D. S. Scott showed that the continuous lattices, invented by him in his study of a mathematical theory of computation [56], are precisely (when they are made into topological spaces via the Scott topology) the injective T0-spaces, i.e., the injective objects in the category T0 of T0-spaces and continuous maps with regard to the class of all embeddings. Moreover, the sort of morphisms between continuous lattices Scott considered are precisely the continuous maps with regard to the respective Scott topologies. These are fairly non-Hausdorff topologies. (Indeed, the Scott topology induces the partial order of the lattice L via x ≦ y if and only if x ∊ cl{j}, the “specialization order” of the topology; hence L is Hausdorff in the Scott topology if and only if L has at most one element.) In topological algebra, compact Lawson semilattices (= compact Hausdorff topological ∧-semilattices such that the ∧-preserving continuous maps into the unit interval, with its ordinary topology and the min-semilattice structure, separate the points) with a unit element 1 have attracted considerable interest.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 5 , 01 October 1985 , pp. 810 - 853
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Alexandrov, P. S., Diskrete Raume, Mat. Sbornik 2 (1937), 501–519.Google Scholar
2. Artim, M., Grothendieck, A. and Verdier, J., Théorie des topos et cohomologie étale des schémas, Springer-Verlag, Lecture Notes in Math. 269 (Berlin-Heidelberg-New York, 1972).Google Scholar
3. Aull, C. E. and Thron, W. J., Separation axioms between T0 and T1 , Indag. Math. 24 (1962), 23–37.Google Scholar
4. Banaschewski, B., Hüllensysteme und Erweiterung von Quasi-Ordnungen, Z. Math. Logik Grundlagen Math. 2 (1956), 117–130.Google Scholar
5. Banaschewski, B., Essential extensions of T0-spaces, General Topology Appl. 7 (1977), 233–246.Google Scholar
6. Banaschewski, B., A lemma on flatness, Algebra Univerlis 12 (1981), 154–159.Google Scholar
7. Banaschewski, B. and Bruns, G., Categorical characterization of the MacNeille completion, Arch. Math. 75 (1967), 369–377.Google Scholar
8. Banaschewski, B. and Hoffmann, R.-E. (editors), Continuous lattices, Proceedings, Bremen (1979). Springer-Verlag, Lecture Notes in Math. 871 (Berlin-Heidelberg-New York, 1981).Google Scholar
9. Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloquium Publications, 3rd ed. (Providence, R.I., 1967).Google Scholar
10. Bruns, G., Darstellungen und Erweiterungen geordneter Mengen, I, II, J. reine angew. Math. 209 (1962), 167–200; 240 (1962), 1–23.Google Scholar
11. Büchi, J. R., Representation of complete lattices by sets, Portugaliae Math. 77 (1952), 154–167.Google Scholar
12. Day, A., Filter monads, continuous lattices and closure systems, Can. J. Math. 27 (1975), 50–59.Google Scholar
13. Dilworth, P. R. and Crawley, P., Decomposition theory for lattices without chain conditions, Trans. Amer. Math. Soc. 96 (1960), 1–22.Google Scholar
14. Egli, H. and Constable, R. L., Computability concepts for programming language semantics, Theoretical Computer Science 2 (1976), 133–145.Google Scholar
15. Erné, M., Completion-invariant extension of the concept of continuous lattices, In: [8], 45–60.Google Scholar
16. Erné, M., On the existence of decompositions in lattices, Algebra Universalis 16 (1983), 338–343.Google Scholar
17. Fell, J. M. G., The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280.Google Scholar
18. Fell, J. M. G., A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472–476.Google Scholar
19. Frink, O., Ideals in partially ordered sets, Amer. Math. Monthly 61 (1954), 223–234.Google Scholar
20. 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).CrossRefGoogle Scholar
21. Gierz, G., Lawson, J. D. and Stralka, A. R., Quasicontinuous posets, Houston J. Math. 9 (1983), 191–208.Google Scholar
22. Hoffmann, R.-E., Charakterisierung nuchterner Raume, Manuscripta Math. 15 (1975), 185–191.Google Scholar
23. Hoffmann, R.-E., Irreducible filters and sober spaces, Manuscripta Math. 22 (1977), 365–380.Google Scholar
24. Hoffmann, R.-E., On the sobrification remainder SX – X, Pacific J. Math. 83 (1979), 145–156.Google Scholar
25. Hoffmann, R.-E., Sobrification of partially ordered sets, Semigroup Forum 17 (1979), 123–138.Google Scholar
26. Hoffmann, R.-E., Essentially complete T0-spaces (I), Manuscripta Math. 27 (1979), 401–432.Google Scholar
27. Hoffmann, R.-E., Projective sober spaces, In: [8], 125–158.Google Scholar
28. Hoffmann, R.-E., Continuous posets and adjoint sequences, Semigroup Forum 18 (1979), 173–188.Google Scholar
29. Hoffmann, R.-E., Topological spaces admitting a “dual”. In: Categorical topology (Springer-Verlag, Lecture Notes in Math. 719 Berlin-Heidelberg-New York, 1979), 157–166.Google Scholar
30. Hoffmann, R.-E., Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications, In: [8], 159–208.Google Scholar
31. Hoffmann, R.-E., Essentially complete T0-spaces, II. A lattice-theoretic approach, Math. Z. 179 (1982), 73–90.Google Scholar
32. Hoffmann, R.-E., Continuous posets: injective hull and MacNeille completion, Preprint.Google Scholar
33. Hoffmann, R.-E., The Fell compactification revisited, Preprint. (Mathematik-Arbeitspapiere 27, Universitàt Bremen, 1982), 68–141. To appear in: Continuous lattices and their applications (Marcel Dekker, Lecture notes in pure and appl. math. 101, New York, 1985), 57–116.Google Scholar
34. Hofmann, K. H., The category DC of completely distributive lattices and their free objects, Seminar on Continuity in Semilattices (SCS), memo, Nov. 24, 1981 (report on a lecture given at the “Miniworkshop” on Continuous Lattices at Tulane University, November 19–21, 1981).Google Scholar
35. Hofmann, K. H., Bernhardina (The essential hull revisited), SCS, memo (1982).Google Scholar
36. Hofmann, K. H. and Lawson, J. D., The spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. 246 (1978), 285–310.Google Scholar
37. Hofmann, K. H. and Mislove, M. W., Local compactness and continuous lattices, In: [8], 209–248.Google Scholar
38. Hofmann, K. H. and Mislove, M. W., A continuous poset whose compactification is not a continuous poset. The square is the injective hull of a discontinuous -compact poset, SCS, memo (1982).Google Scholar
39. Hofmann, K. H., Mislove, M. and Stralka, A. R., The Pontryagin duality of compact 0-dimensional semilattices and its applications, Springer-Verlag, Lecture Notes in Math. 396 (Berlin-Heidelberg-New York, 1974).CrossRefGoogle Scholar
40. Hofmann, K. H. and Stralka, A. R., The algebraic theory of Lawson semilattices — Applications of Galois connections to compact semilattices, Dissertationes Math. (– Rozprawy Mat.) 137, Warsaw (1976), 1–54.Google Scholar
41. Isbell, J. R., Meet-continuous lattices, Symposia Mathematica 16 (Convegno sulla Topologica Insiemsistica e générale, INDAM, Roma, Marzo 1973), 41–54. (Academic Press, London, 1975.)Google Scholar
42. Isbell, J. R., Completion of a construction of Johnstone, Proc. Amer. Math. Soc. 85 (1982), 333–334.Google Scholar
43. Johnstone, P. T., Scott is not always sober, In: [8], 282–283.Google Scholar
44. Kamara, M., Treillis continus et treillis complètement distributifs, Semigroup Forum 16 (1978), 387–388.Google Scholar
45. Lawson, J. D., Intrinsic topologies in topological lattices and semilattices, Pacific J. Math. 44 (1973), 593–602.Google Scholar
46. Lawson, J. D., The duality of continuous posets, Houston J. Math. 5 (1979), 357–394.Google Scholar
47. Lea, J. W. Jr., Continuous lattices and compact Lawson semilattices, Semigroup Forum 13 (1976), 387–388.Google Scholar
48. Linton, F. E. J., An outline offunctorial semantics. In: Seminar on Triples and Categorical Homology Theory (Springer-Verlag, Lecture Notes in Math., 80, 1969), 7–52.Google Scholar
49. Markowsky, G., A motivation and generalization of Scott's notion of a continuous lattice, Preprint (1977). Revised version in: [8], 298–307.Google Scholar
50. Markowsky, G. and Rosen, B. K., Bases for chain-complete posets, IBM J. Res. Development 20 (1976), 137–147.Google Scholar
51. MacNeille, H., Partially ordered sets, Trans. Amer. Math. Soc. 42 (1937), 416–460.Google Scholar
52. Nachbin, L., Topology and order (Van Nostrand, Princeton, 1965).Google Scholar
53. Nel, L. D. and Wilson, R. G., Epireflect ions in the category of T0-spaces, Fund. Math. 75 (1972), 69–74.Google Scholar
54. Plotkin, G. D., A power domain construction, SIAM J. Computing 5 (1976), 452–487.Google Scholar
55. Raney, G. N., Completely distributive complete lattices, Proc. Amer. Math. Soc. 3 (1952), 677–680.Google Scholar
56. Scott, D., Outline of a mathematical theory of computation, Proc. 4th Ann. Princeton Conf. on Information Science and Systems (1970), 169–176.Google Scholar
57. Raney, G. N., Continuous lattices, In: Toposes, algebraic geometry and logic, Proc. conference, Halifax (1971), 97–136 (Springer-Verlag, Lecture Notes in Math. 274).Google Scholar
58. Skula, L., On a reflective subcategory of the category of all topological spaces, Trans. Amer. Math. Soc. 142 (1969), 37–41.Google Scholar
59. Smyth, M. B., Effectively given domains, Theoretical Computer Science 5 (1977), 257–274.Google Scholar
60. Thron, W. J., Lattice-equivalence of topological spaces, Duke Math. J. 29 (1962), 671–679.Google Scholar
61. Wilson, R. L., Relationships between continuous posets and compact Lawson posets, Notices Amer. Math. Soc. 24 (1977).Google Scholar
62. Wilson, R. L., Intrinsic topologies on partially ordered sets and results on compact semigroups, Ph.D. thesis, University of Tennessee (1978).Google Scholar
63. Wyler, O., Dedekind complete posets and Scott topologies (SCS, memo, 1977). Revised version in: [8], 384–389.Google Scholar