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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 64, Issue 3


Extending semilattices to frames using sites and coverages

Richard Ball / Aleš Pultr
  • Department of Applied Mathematics and CE-ITI, MFF, Charles University, CZ-11800, Praha 1 Malostranské Nám., Czech Reublic
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Published Online: 2014-07-05 | DOI: https://doi.org/10.2478/s12175-014-0223-9


Each meet semilattice S is well known to be freely extended to a frame by its down-sets DS. In this article we present, first, the complete range of frame extensions generated by S; it turns out to be a sub-coframe of the coframe C of sublocales of DS, indeed, an interval in C, with DS as the top and the extension of S respecting all the exact joins in S as the bottom. Then, the Heyting and Boolean case is discussed; there, the bottom extension is shown to coincide with the Dedekind-MacNeille completion. The technique used is a technique of sites, generalizing that used in [JOHNSTONE, P. T.: Stone Spaces. Cambridge Stud. Adv. Math. 3, Cambridge University Press, Cambridge, 1982].

MSC: Primary 05C38, 15A15; Secondary 05A15, 15A18

Keywords: site; coverage; frame; meet semilattice; joins in meet semilattices

  • [1] BALL, R. N.: Distributive Cauchy lattices, Algebra Universalis 18 (1984), 134–174. http://dx.doi.org/10.1007/BF01198525CrossrefGoogle Scholar

  • [2] BALL, R. N.— PULTR, A.: Quotients and colimits of κ-quantales, Topology Appl. 158 (2011), 2294–2306. http://dx.doi.org/10.1016/j.topol.2011.06.029CrossrefWeb of ScienceGoogle Scholar

  • [3] BALL, R. N.— PULTR, A.— PICADO, J.: Notes on exact meets and joins, Appl. Categ. Structures (To appear) Web of ScienceGoogle Scholar

  • [4] BANASCHEWSKI, B.— PULTR, A.: Scott information systems, frames, and domains, Mathematik-Arbeitspapiere (Universität Bremen) 54 (2000), 35–46. Google Scholar

  • [5] BRUNS, G.— LAKSER, H.: Injective hulls of semilattices, Canad. Math. Bull. 13 (1970), 115–118. http://dx.doi.org/10.4153/CMB-1970-023-6CrossrefGoogle Scholar

  • [6] JOHNSTONE, P. T.: Stone Spaces. Cambridge Stud. Adv. Math. 3, Cambridge University Press, Cambridge, 1982. Google Scholar

  • [7] PICADO, J.— PULTR, A.: Locales Treated Mostly in a Covariant Way. Textos Mat. Sér. B 41, Univ. Coimbra, Coimbra, 2008. Google Scholar

  • [8] PICADO, J.— PULTR, A.: Frames and Locales: Topology without Points, Front. Math. 28, Springer, Basel, 2012. http://dx.doi.org/10.1007/978-3-0348-0154-6CrossrefGoogle Scholar

  • [9] PULTR, A.: Frames. In: Handbook of Algebra, Vol. 3 (M. Hazewinkel, ed.), Elsevier, Amsterdam, 2003. Google Scholar

  • [10] WIGNER, D.: Two notes on frames, J. Aust. Math. Soc. Ser. A 28 (1979), 257–268. http://dx.doi.org/10.1017/S1446788700012209CrossrefGoogle Scholar

About the article

Published Online: 2014-07-05

Published in Print: 2014-06-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 3, Pages 527–544, ISSN (Online) 1337-2211, DOI: https://doi.org/10.2478/s12175-014-0223-9.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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