Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-01T04:29:01.307Z Has data issue: false hasContentIssue false
  • English
  • Français

The Duality of Distributive Continuous Lattices

Published online by Cambridge University Press:  20 November 2018

B. Banaschewski
B. Banaschewski*
Affiliation:
McMaster University, Hamilton, Ontario
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.

Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 385 - 394
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Banaschewski, B., Essential extensions of To-spaces, Gen. Top. Appl. 7 (1977), 233–246.Google Scholar
2. Benabou, J., Treillis locaux et par atop ologies, Sém. C. Ehresmann 1957/58. Fac. de Sci. Paris, 1959.Google Scholar
3. Birkhoff, G., Lattice theory, Amer. Math. Soc. Coll. Publ. 25 (American Mathematical Society, Providence, Rhode Island, 1967).Google Scholar
4. Day, R. A., Filter monads, continuous lattices, and closure systems, Can. J. Math. 27 (1975), 50–59.Google Scholar
5. Dowker, C. and Papert, D., Quotient frames and subspaces, Proc. London Math. Soc. 16 (1966), 275–296.Google Scholar
6. Hochster, M., Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60.Google Scholar
7. Hofmann, K. H. and Lawson, J. D., The spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. (to appear).Google Scholar
8. Isbell, J. R., Atomless parts of spaces, Math. Scand. 31 (1972), 5–32.Google Scholar
9. Linton, F. E. J., Some aspects of equational theories, Proc. Conference on Categorical Algebra La Jolla, 1965 (Springer-Verlag, Berlin, Heidelberg, New York, 1966), 84–94.Google Scholar
10. MacLane, S., Categories for the working mathematician, Graduate Texts in Mathematics 5 (Springer-Verlag, New York, Heidelberg, Berlin, 1971).Google Scholar
11. Scott, D., Continuous lattices, Lecture Notes in Mathematics 274 (Springer-Verlag, Berlin, Heidelberg, New York, 1972), 97–136.Google Scholar
12. Stone, M. H., Topological representation of distributive lattices and Browerian logics, Cas. Mat. Fys. 67 (1937), 1–25.Google Scholar