Bitopological duality for distributive lattices and Heyting algebras

Mathematical Structures in Computer Science

Volumn 20, Issue 3, 2010, Pages 359-393

  Bezhanishvili, Guram ^a   Bezhanishvili, Nick ^b   Gabelaia, David ^c   Kurz, Alexander ^d  

^a NEW MEXICO STATE UNIVERSITY   (United States)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

^c A RAZMADZE MATHEMATICAL INSTITUTE   (Georgia)

^d UNIVERSITY OF LEICESTER   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDED DISTRIBUTIVE LATTICE; DISTRIBUTIVE LATTICE; GENERALISATION; HEYTING ALGEBRAS; PRIESTLEY SPACE; SPECTRAL SPACES; STONE SPACES;

ALGEBRA;

EID: 77957301539     PISSN: 09601295     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0960129509990302     Document Type: Article

References (33)

1. Abramsky, S. (1991) Domain theory in logical form. Ann. Pure Appl. Logic 51 (1-2) 1-77.
2. Adams, M. E. (1973) The Frattini sublattice of a distributive lattice. Algebra Universalis 3 216-228.
3. Alvarez-Manilla, M., Jung, A. and Keimel, K. (2004) The probabilistic powerdomain for stably compact spaces. Theoretical Computer Science 328(3) 221-244. (Pubitemid 41155305)
4. Bonsangue, M. and Kurz, A. (2007) Pi-calculus in logical form. In: 22nd IEEE Symposium on Logic in Computer Science (LICS 2007), IEEE 303-312.
5. Cignoli, R., Lafalce, S. and Petrovich, A. (1991) Remarks on Priestley duality for distributive lattices. Order 8(3) 299-315.
6. Cornish, W. H. (1975) On H. Priestley's dual of the category of bounded distributive lattices. Mat. Vesnik 12(27) (4) 329-332.
7. Dvalishvili, B. P. (2005) Bitopological spaces: theory, relations with generalized algebraic structures, and applications. North-Holland Mathematics Studies 199.
8. Esakia, L. L. (1974) Topological Kripke models. Soviet Math. Dokl. 15 147-151.
9. Esakia, L. L. (1975) The problem of dualism in the intuitionistic logic and Browerian lattices. In: V Inter. Congress of Logic, Methodology and Philosophy of Science 7-8.
10. Esakia, L. L. (1985) Heyting Algebras I. Duality Theory (in Russian), 'Metsniereba', Tbilisi.
11. Fleisher, I. (2000) Priestley's duality from Stone's. Adv. in Appl. Math. 25(3) 233-238.
12. Gehrke, M., Grigorieff, S. and Pin, J.-E. (2008) Duality and equational theory of regular languages. In: Aceto, L., Damgaard, I., Goldberg, L., Halldorsson, M., Ingolfsdottir, A. and Walukiewicz, I. (eds.) Automata, Languages and Programming. Springer-Verlag Lecture Notes in Computer Science 5125 246-257.
13. Gehrke, M. and Jónsson, B. (1994) Bounded distributive lattices with operators. Math. Japon. 40(2) 207-215.
14. Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous lattices and domains. Encyclopedia of Mathematics and its Applications 93, Cambridge University Press.
15. Harding, J. and Bezhanishvili, G. (2004) MacNeille completions of Heyting algebras. Houston Journal of Mathematics 30 937-952. (Pubitemid 40031032)
16. Hochster, M. (1969) Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 43-60.
17. Johnstone, P. T. (1982) Stone spaces, Cambridge University Press.
18. Jung, A., Kegelmann, M. and Moshier, M. A. (1997) Multilingual sequent calculus and coherent spaces. In: Mathematical foundations of programming semantics (Pittsburgh, PA, 1997). Electronic Notes in Theoretical Computer Science 6 203-220. (Pubitemid 129526929)
19. Jung, A., Kegelmann, M. and Moshier, M. A. (2001) Stably compact spaces and closed relations. In: Brookes, S. and Mislove, M. (eds.) 17th Conference on Mathematical Foundations of Programming Semantics Electronic Notes in Theoretical Computer Science 45 209-231. (Pubitemid 40701590)
20. Jung, A. and Moshier, M. A. (2006) On the bitopological nature of Stone duality. Technical Report CSR-06-13, School of Computer Science, University of Birmingham.
21. Jung, A. and Sünderhauf, P. (1996) On the duality of compact vs. open. In: Papers on general topology and applications (Gorham, ME, 1995). Ann. New York Acad. Sci. 806 214-230. (Pubitemid 126499781)
22. Kelly, J. C. (1963) Bitopological spaces. Proc. London Math. Soc. s3-13 71-89.
23. Kopperman, R. (1995) Asymmetry and duality in topology. Topology Appl. 66(1) 1-39.
24. Lawson, J. D. (1991) Order and strongly sober compactifications. In: Topology and category theory in computer science (Oxford, 1989), Oxford University Press 179-205.
25. Priestley, H. A. (1970) Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2 186-190.
26. Priestley, H. A. (1972) Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. s3-24 507-530.
27. Priestley, H. A. (1984) Ordered sets and duality for distributive lattices. In: Orders: description and roles (L'Arbresle, 1982). North-Holland Math. Stud. 99 39-60.
28. Reilly, I. L. (1973) Zero dimensional bitopological spaces. Indag. Math. 35 127-131.
29. Salbany, S. (1974) Bitopological spaces, compactifications and completions. Mathematical Monographs of the University of Cape Town 1.
30. Salbany, S. (1984) A bitopological view of topology and order. In: Categorical topology (Toledo, Ohio, 1983). Sigma Ser. Pure Math. 5 481-504.
31. Schmid, J. (2002) Quasiorders and sublattices of distributive lattices. Order 19(1) 11-34. (Pubitemid 37134191)
32. Smyth, M. B. (1992) Stable compactification. I. J. London Math. Soc. s2-45 (2) 321-340.
33. Stone, M. (1937) Topological representation of distributive lattices and Brouwerian logics. Časopis Pešt. Mat. Fys. 67 1-25.