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Documents authored by Kovar, Martin

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Kovar, Martin

Document
DOI: 10.4230/DagSemProc.04351.19
The de Groot dual for general collections of sets

Authors: Martin Kovar

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract
A topology is de Groot dual of another topology, if it has a closed base consisting of all its compact saturated sets. Until 2001 it was an unsolved problem of J. Lawson and M. Mislove whether the sequence of iterated dualizations of a topological space is finite. In this paper we generalize the author's original construction to an arbitrary family instead of a topology. Among other results we prove that for any family $\C\subseteq 2^X$ it holds $\C^{dd}=\C^{dddd}$. We also show similar identities for some other similar and topology-related structures.
Cite as

Martin Kovar. The de Groot dual for general collections of sets. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.19,
  author =	{Kovar, Martin},
  title =	{{The de Groot dual for general collections of sets}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--8},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.19},
  URN =		{urn:nbn:de:0030-drops-1215},
  doi =		{10.4230/DagSemProc.04351.19},
  annote =	{Keywords: Saturated set , dual topology , compactness operator}
}
Document
DOI: 10.4230/DagSemProc.04351.20
The Hofmann-Mislove Theorem for general posets

Authors: Martin Kovar

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract
In this paper we attempt to find and investigate the most general class of posets which satisfy a properly generalized version of the Hofmann-Mislove theorem. For that purpose, we generalize and study some notions (like compactness, the Scott topology, Scott open filters, prime elements, the spectrum etc.), and adjust them for use in general posets. Then we characterize the posets satisfying the Hofmann-Mislove theorem by the relationship between the generalized Scott closed prime subsets and the generalized prime elements of the poset. The theory become classic for distributive lattices. Remark that the topologies induced on the generalized spectra in general need not be sober.
Cite as

Martin Kovar. The Hofmann-Mislove Theorem for general posets. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.20,
  author =	{Kovar, Martin},
  title =	{{The Hofmann-Mislove Theorem for general posets}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--16},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.20},
  URN =		{urn:nbn:de:0030-drops-1199},
  doi =		{10.4230/DagSemProc.04351.20},
  annote =	{Keywords: Posets , generalized Scott topology , Scott open filters , (filtered) compactness , saturated}
}
Document
DOI: 10.4230/DagSemProc.04351.21
The Hofmann-Mislove Theorem for general topological structures

Authors: Martin Kovar

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract
In this paper we prove a modification of Hofmann-Mislove theorem for a topological structure similar to the minusspaces of de Groot, in which the empty set "need not be open". This will extend, in a slightly relaxed form, the validity of the classical Hofmann-Mislove theorem also to some of those spaces, whose underlying topology need not be (quasi-) sober.
Cite as

Martin Kovar. The Hofmann-Mislove Theorem for general topological structures. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.21,
  author =	{Kovar, Martin},
  title =	{{The Hofmann-Mislove Theorem for general topological structures}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.21},
  URN =		{urn:nbn:de:0030-drops-1208},
  doi =		{10.4230/DagSemProc.04351.21},
  annote =	{Keywords: Compact saturated set , Scott open filter , (quasi-) sober space}
}
Document
DOI: 10.4230/DagSemProc.04351.17
On Maximality of Compact Topologies

Authors: Martin Kovar

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract
Using some advanced properties of the de Groot dual and some generalization of the Hofmann-Mislove theorem, we solve in the positive the question of D. E. Cameron: Is every compact topology contained in some maximal compact topology?
Cite as

Martin Kovar. On Maximality of Compact Topologies. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.17,
  author =	{Kovar, Martin},
  title =	{{On Maximality of Compact Topologies}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.17},
  URN =		{urn:nbn:de:0030-drops-1182},
  doi =		{10.4230/DagSemProc.04351.17},
  annote =	{Keywords: de Groot dual , compact saturated set , wide Scott open filter , maximal compact topology}
}

Kovár, Martin Maria

Document
DOI: 10.4230/DagSemProc.06341.6
Instant topological relationships hidden in the reality

Authors: Martin Maria Kovár

Published in: Dagstuhl Seminar Proceedings, Volume 6341, Computational Structures for Modelling Space, Time and Causality (2007)

Abstract
In most applications of general topology, topology usually is not the first, primary structure, but the information which finally leads to the construction of the certain, for some purpose required topology, is filtered by more or less thick filter of the other mathematical structures. This fact has two main consequences: (1) Most important applied constructions may be done in the primary structure, bypassing the topology. (2) Some topologically important information from the reality may be lost (filtered out by the other, front-end mathematical structures). Thus some natural and direct connection between topology and the reality could be useful. In this contribution we will discuss a pointless topological structure which directly reflects relationship between various locations which are glued together by possible presence of a physical object or a virtual ``observer".
Cite as

Martin Maria Kovár. Instant topological relationships hidden in the reality. In Computational Structures for Modelling Space, Time and Causality. Dagstuhl Seminar Proceedings, Volume 6341, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{kovar:DagSemProc.06341.6,
  author =	{Kov\'{a}r, Martin Maria},
  title =	{{Instant topological relationships hidden in the reality}},
  booktitle =	{Computational Structures for Modelling Space, Time and Causality},
  pages =	{1--2},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{6341},
  editor =	{Ralph Kopperman and Prakash Panangaden and Michael B. Smyth and Dieter Spreen},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06341.6},
  URN =		{urn:nbn:de:0030-drops-8962},
  doi =		{10.4230/DagSemProc.06341.6},
  annote =	{Keywords: Pointless topology, reality}
}
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