Local Compactness in MT-Algebras
Keywords:
compactness, duality theory, interior operator, local compactness, pointfree topologyAbstract
In our previous work, we introduced McKinsey-Tarski algebras (MT-algebras for short) as an alternative pointfree approach to topology. Here, we study local compactness in MT-algebras. We establish the Hofmann-Mislove theorem for sober MT-algebras, which we use to develop the MT-algebra versions of such well-known dualities in pointfree topology as Hofmann-Lawson, Isbell, and Stone dualities. This yields a new perspective on these classic results.
References
Raymond Balbes and Philip Dwinger, Distributive Lattices. Columbia, MO: University of Missouri Press, 1974.
B. Banaschewski, The duality of distributive continuous lattices, Canadian J. Math. 32 (1980), no. 2, 385-394.
Bernhard Banaschewski, Coherent frames in Continuous Lattices. Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices (Workshop IV) (University of Bremen, Bremen, November 9-11, 1979). Ed. Bernhard Banaschewski and Rudolf-Eberhard Hoffmann. Lecture Notes in Mathematics, 871. Berlin-New York: Springer-Verlag, 1981. 1-11.
B. Banaschewski, Universal zero-dimensional compactifications in Categorical Topology and Its Relation to Analysis, Algebra and Combinatorics: Prague, Czechoslovakia, 22-27 August 1988. Ed. Jiří Adámek and Saunders Mac Lane. Teaneck, NJ: World Scientific Publishing Co., Inc., 1989. 257-269.
B. Banaschewski and G. C. L. Brümmer, Stably continuous frames, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 7-19.
B. Banaschewski and C. J. Mulvey, Stone-Cech compactification of locales. I, Houston J. Math. 6 (1980), no. 3, 301-312.
G. Bezhanishvili, L. Carai, and P. Morandi, A point-free approach to canonical extensions of Boolean algebras and bounded Archimedean $\ell$-algebras, Order 40 (2023), no. 2, 257-287.
Guram Bezhanishvili and Andre Kornell, On the structure of modal and tense operators on a boolean algebra. Available at https://arxiv.org/abs/2308.08664.
Guram Bezhanishvili and Sebastian Melzer, Hofmann-Mislove through the lenses of Priestley, Semigroup Forum 105 (2022), no. 3, 825-833.
Guram Bezhanishvili, Ray Mines, and Patrick J. Morandi, Topo-canonical completions of closure algebras and Heyting algebras, Algebra Universalis 58 (2008), no. 1, 1-34.
Guram Bezhanishvili and Ranjitha Raviprakash, McKinsey-Tarski algebras: An alternative pointfree approach to topology, Topology Appl. 339 (2023), Paper No. 108689, 30 pp.
Ryszard Engelking, General Topology. Translated from the Polish by the author. 2nd ed. Sigma Series in Pure Mathematics, 6. Berlin: Heldermann Verlag, 1989.
Marcel Erné, The strength of prime separation, sobriety, and compactness theorems, Topology Appl. 241 (2018), 263-290.
Leo Esakia, Heyting Algebras: Duality Theory. Ed. Guram Bezhanishvili and Wesley H. Holliday. Translated from the Russian edition by Anton Evseev. Trends in Logic-Studia Logica Library, 50. Cham: Springer, 2019.
Mai Gehrke and John Harding, Bounded lattice expansions, J. Algebra 238 (2001), no. 1,345-371.
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, 93. Cambridge: Cambridge University Press, 2003.
Gerhard Gierz and Klaus Keimel, A lemma on primes appearing in algebra and analysis, Houston J. Math. 3 (1977), no. 2, 207-224.
Karl H. Hofmann and Jimmie D. Lawson, The spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. 246 (1978), 285-310.
Karl H. Hofmann and Michael W. Mislove, Local compactness and continuous lattices in Continuous Lattices. Ed. Bernhard Banaschewski and Rudolf- Eberhard Hoffmann. Lecture Notes in Mathematics, 871. Berlin-New York: Springer-Verlag, 1981. 209-248.
Wesley H. Holliday, Possibility frames and forcing for modal logic. Available at https://escholarship.org/uc/item/5462j5b6.
John R. Isbell, Atomless parts of spaces, Math. Sc and. 31 (1972), 5-32.
P. T. Johnstone, The Gleason cover of a topos. I, J. Pure Appl. Algebra 19 (1980), 171-192.
Peter T. Johnstone, Stone Spaces. Cambridge Studies in Advanced Mathematics, 3. Cambridge: Cambridge University Press, 1986.
Bjarni Jónsson and Alfred Tarski, Boolean algebras with operators. I, Amer. J. Math. 73 (1951), 891-939.
Klaus Keimel and Jan Paseka, A direct proof of the Hofmann-Mislove theorem, Proc. Amer. Math. Soc. 120 (1994), no. 1, 301-303.
K. Kuratowski, Sur l'Opération À de l'Analysis Situs, Fund. Math., 3 (1922), 182-199.
J. C. C. McKinsey and Alfred Tarski, The algebra of topology, Ann. of Math. (2) 45 (1944), 141-191.
Colin Naturman, Interior Algebras and Topology. PhD thesis, University of Cape Town, 1991.
Jorge Picado and Aleš Pultr, Frames and Locales: Topology Without Points. Frontiers in Mathematics. Basel: Birkhäuser/Springer Basel AG, 2012.
Jorge Picado and Aleš Pultr, Separation in Point-Free Topology. Cham: Birkhäuser/Springer, 2021.
Helena Rasiowa and Roman Sikorski, The Mathematics of Metamathematics. Monografie Matematyczne, Tom 41 [Mathematical Monographs]. Warsaw: Państwowe Wydawnictwo Naukowe, 1970.
H. Simmons, A couple of triples, Topology Appl. 13 (1982), no. 2, 201-223.
M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), no. 1, 37-111.
Steven Vickers, Topology Via Logic. Cambridge Tracts in Theoretical Computer Science, 5. Cambridge: Cambridge University Press, 1989.
