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Mathematica Slovaca

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Volume 68, Issue 2

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Compactifications of partial frames via strongly regular ideals

John Frith
  • Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, Cape Town, South Africa
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/ Anneliese Schauerte
  • Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, Cape Town, South Africa
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Published Online: 2018-03-31 | DOI: https://doi.org/10.1515/ms-2017-0100

Abstract

Partial frames provide a fertile context in which to do pointfree structured and unstructured topology, using a small collection of axioms of an elementary nature. Amongst other things they can be used to investigate similarities and differences between frames, σ-frames and κ-frames.

In this paper, the theory of strong inclusions for partial frames is used to describe compactifications of completely regular partial frames; the elements of these compactifications are given explicitly as strongly regular ideals. This is independent of and encompasses the theory of compactifications for frames. As an application, we revisit the Samuel compactification of a uniform partial frame.

MSC 2010: Primary 06D22; Secondary 06B10; 54D35; 54E15

Keywords: frame; partial frame; 𝓢-frame; κ-frame; σ-frame; updirected; strongly regular ideal; compactification; strong inclusion; Samuel compactification

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About the article


Received: 2015-12-07

Accepted: 2016-06-08

Published Online: 2018-03-31

Published in Print: 2018-04-25


Citation Information: Mathematica Slovaca, Volume 68, Issue 2, Pages 285–298, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0100.

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