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DUALITY FOR CONVEX POLYTOPES

Part of: General convexity Algebraic structures General theory of categories and functors Categories and theories

Published online by Cambridge University Press:  01 June 2009

A. B. ROMANOWSKA ,
P. ŚLUSARSKI  and
J. D. H. SMITH
A. B. ROMANOWSKA*
Affiliation:
Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warsaw, Poland (email: aroman@mini.pw.edu.pl)
P. ŚLUSARSKI
Affiliation:
Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warsaw, Poland (email: P.Slusarski@mini.pw.edu.pl)
J. D. H. SMITH
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA (email: jdhsmith@math.iastate.edu)
*
✉For correspondence; e-mail: aroman@mini.pw.edu.pl
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Abstract

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This paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations.

Keywords

dualitymodesaffine spacesbarycentric algebrasconvex setssimplexhypercube

MSC classification

Secondary: 18C05: Equational categories 52A01: Axiomatic and generalized convexity 08A05: Structure theory 18A35: Categories admitting limits (complete categories), functors preserving limits, completions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 3 , June 2009 , pp. 399 - 412
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This paper was written within the framework of INTAS project no. 03-51-4110.

References

[1]Brøndsted, A., An Introduction to Convex Polytopes (Springer, New York, 1983).CrossRefGoogle Scholar
[2]Clark, D. M. and Davey, B. A., Natural Dualities for the Working Algebraist (Cambridge University Press, Cambridge, 1998).Google Scholar
[3]Davey, B. A., ‘Duality theory on ten dollars a day’, in: Algebras and Orders (eds. I. G. Rosenberg and G. Sabidussi) (Kluwer Academic, Dordrecht, 1993), pp. 71–111.Google Scholar
[4]Davey, B. A. and Werner, H., ‘Dualities and equivalences for varieties of algebras’, Colloq. Math. Soc. János Bolyai 33 (1983), 101–275.Google Scholar
[5]Grünbaum, B., Convex Polytopes, 2nd edn (Springer, New York, 2003).CrossRefGoogle Scholar
[6]Hofmann, K. H., Mislove, M. and Stralka, A., Pontryagin Duality of Compact 0-Dimensional Semilattices and its Applications, Lecture Notes in Mathematics, 396 (Springer, Berlin, 1974).CrossRefGoogle Scholar
[7]Johnstone, P. T., Stone Spaces (Cambridge University Press, Cambridge, 1982).Google Scholar
[8]Mac Lane, S., Categories for the Working Mathematician (Springer, Berlin, 1971).Google Scholar
[9]Neumann, W. D., ‘On the quasivariety of convex subsets of affine spaces’, Arch. Math. 21 (1970), 11–16.CrossRefGoogle Scholar
[10]Pszczoła, K. J., ‘Duality for affine spaces over finite fields’, Contributions to General Algebra 13 (2001), 285–293.Google Scholar
[11]Pszczoła, K. J., Romanowska, A. B. and Smith, J. D. H., ‘Duality for some free modes’, Discuss. Math. Gen. Algebra Appl. 23 (2003), 45–61.CrossRefGoogle Scholar
[12]Pszczoła, K. J., Romanowska, A. B. and Smith, J. D. H., ‘Duality for quadrilaterals’, Contributions to General Algebra 14 (2004), 127–134.Google Scholar
[13]Romanowska, A. B. and Smith, J. D. H., Modal Theory (Heldermann, Berlin, 1985).Google Scholar
[14]Romanowska, A. B. and Smith, J. D. H., Modes (World Scientific, Singapore, 2002).CrossRefGoogle Scholar
[15]Smith, J. D. H. and Romanowska, A. B., Post-modern Algebra (Wiley, New York, 1999).CrossRefGoogle Scholar