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DUALITY FOR QUASIPOLYTOPES

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

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. 

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