License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.04351.4
URN: urn:nbn:de:0030-drops-1253
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2005/125/
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Pfaltz, John L.

A Category of Discrete Closure Spaces

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04351.PfaltzJohn1.Paper.125.pdf (0.2 MB)

Abstract

Discrete systems such as sets, monoids, groups are familiar categories.
The internal strucutre of the latter two is defined by an algebraic operator.
In this paper we describe the internal structure of the base set by a closure operator. We illustrate the role of such closure in convex geometries and partially ordered sets and thus suggestthe wide applicability of closure systems.
Next we develop the ideas of closed and complete functions over closure spaces. These can be used to establish criteria for asserting
that "the closure of a functional image under $f$ is equal to the functional image of the closure". Functions with these properties can be treated as categorical morphisms. Finally, the category "CSystem" of closure systems is shown to be cartesian closed.

BibTeX - Entry

@InProceedings{pfaltz:DagSemProc.04351.4,
  author =	{Pfaltz, John L.},
  title =	{{A Category of Discrete Closure Spaces}},
  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/opus/volltexte/2005/125},
  URN =		{urn:nbn:de:0030-drops-1253},
  doi =		{10.4230/DagSemProc.04351.4},
  annote =	{Keywords: Category , closure , antimatroid , function}
}

Keywords: Category , closure , antimatroid , function
Collection: 04351 - Spatial Representation: Discrete vs. Continuous Computational Models
Issue Date: 2005
Date of publication: 22.04.2005

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