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DUALITY FOR QUASIPOLYTOPES

Part of: Other classes of algebra Algebraic structures General convexity Polytopes and polyhedra

Published online by Cambridge University Press:  26 February 2016

A. MUĆKA  and
A. B. ROMANOWSKA
A. MUĆKA*
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warsaw, Poland email [email protected]
A. B. ROMANOWSKA
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warsaw, Poland email [email protected]
*
✉[email protected]
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Abstract

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In an earlier paper, Romanowska, Ślusarski and Smith described 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. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.

Keywords

modeaffine spaceconvex setbarycentric algebrasemilatticevariety regularizationPłonka sumduality

MSC classification

Primary: 08A05: Structure theory
Secondary: 08C05: Categories of algebras 08C99: None of the above, but in this section 52B11: $n$-dimensional polytopes 52A01: Axiomatic and generalized convexity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 1 , August 2016 , pp. 95 - 117
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Bergman, C. and Romanowska, A., ‘Subquasivarieties of regularized varieties’, Algebra Universalis 36 (1996), 536–563.CrossRefGoogle Scholar
Brøndsted, A., An Introduction to Convex Polytopes (Springer, New York, 1983).CrossRefGoogle Scholar
Clark, D. M. and Davey, B. A., Natural Dualities for the Working Algebraist (Cambridge University Press, Cambridge, 1998).Google Scholar
Csákány, B., ‘Varieties of affine modules’, Acta Sci. Math. 37 (1975), 3–10.Google Scholar
Czédli, G. and Romanowska, A., ‘Generalized convexity and closure conditions’, Algebra Universalis 69 (2013), 171–190.Google Scholar
Davey, B. A., ‘Duality theory on ten dollars a day’, in: Algebras and Orders (eds. Rosenberg, I. G. and Sabidussi, G.) (Kluwer Academic, 1993), 71–111.Google Scholar
Davey, B. A. and Knox, B. J., ‘Regularising natural dualities’, Acta Math. Univ. Comenian. LXVIII (1999), 295–318.Google Scholar
Davey, B. A. and Werner, H., ‘Dualities and equivalences for varieties of algebras’, Colloq. Math. Soc. János Bolyai 33 (1983), 101–275.Google Scholar
Gierz, G. and Romanowska, A., ‘Duality for distributive bisemilattices’, J. Aust. Math. Soc. 51 (1991), 247–275.CrossRefGoogle Scholar
Grünbaum, B., Convex Polytopes, 2nd edn (Springer, New York, 2003).CrossRefGoogle Scholar
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).Google Scholar
Ignatov, V. V., ‘Quasivarieties of convexors’, Izv. Vyssh. Uchebn. Zaved. Mat. 29 (1985), 12–14 (in Russian).Google Scholar
Johnstone, P. T., Stone Spaces (Cambridge University Press, Cambridge, 1982).Google Scholar
Mac Lane, S., Categories for the Working Mathematician (Springer, Berlin, 1971).Google Scholar
Neumann, W. D., ‘On the quasivariety of convex subsets of affine spaces’, Arch. Math. 21 (1970), 11–16.Google Scholar
Płonka, J., ‘On the sum of a direct system of universal algebras with nullary polynomials’, Algebra Universalis 19 (1984), 197–207.Google Scholar
Płonka, J. and Romanowska, A., ‘Semilattice sums’, in: Universal Algebra and Quasigroup Theory (eds. Romanowska, A. and Smith, J. D. H.) (Heldermann, Berlin, 1992), 123–158.Google Scholar
Pszczoła, K. J., ‘Duality for affine spaces over finite fields’, Contrib. Gen. Algebra 13 (2001), 285–293.Google Scholar
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
Pszczoła, K. J., Romanowska, A. B. and Smith, J. D. H., ‘Duality for quadrilaterals’, Contrib. Gen. Algebra 14 (2004), 127–134.Google Scholar
Romanowska, A. B., Ślusarski, P. and Smith, J. D. H., ‘Duality for convex polytopes’, J. Aust. Math. Soc. 86 (2009), 399–412.CrossRefGoogle Scholar
Romanowska, A. B. and Smith, J. D. H., Modal Theory (Heldermann, Berlin, 1985).Google Scholar
Romanowska, A. B. and Smith, J. D. H., ‘On the structure of barycentric algebras’, Houston J. Math. 16 (1990), 431–448.Google Scholar
Romanowska, A. B. and Smith, J. D. H., ‘Semilattice-based dualities’, Studia Logica 56 (1996), 225–261.Google Scholar
Romanowska, A. B. and Smith, J. D. H., ‘Duality for semilattice representations’, J. Pure Appl. Algebra 115 (1997), 289–308.CrossRefGoogle Scholar
Romanowska, A. B. and Smith, J. D. H., Modes (World Scientific, Singapore, 2002).Google Scholar
Romanowska, A. B., Smith, J. D. H. and Orłowska, E., ‘Abstract barycentric algebras’, Fund. Inform. 81 (2007), 1–17.Google Scholar
Ślusarski, P., ‘Duality for some classes of convex sets’, PhD Thesis, Faculty of Mathematics and Information Science, Warsaw University of Technology, 2007.Google Scholar
Smith, J. D. H. and Romanowska, A. B., Post-Modern Algebra (Wiley, New York, 1999).Google Scholar