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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


IMPACT FACTOR 2018: 0.490

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Volume 64, Issue 3

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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

Abstract

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

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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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R.N. Ball, M. A. Moshier, J.L. Walters-Wayland, and A. Pultr
Quaestiones Mathematicae, 2017, Volume 40, Number 3, Page 347

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