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2017 | OriginalPaper | Chapter

9. Generalised Heaps as Affine Structures

Authors : Christopher D. Hollings, Mark V. Lawson

Published in: Wagner’s Theory of Generalised Heaps

Publisher: Springer International Publishing

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Abstract

We describe how Wagner’s notion of a generalised heap can be transformed into something much more akin to a generalised bitorsor with the help of Anders Kock’s concept of a pregroupoid. This directly generalises the classical relationship between heaps and bitorsors of groups. The generalised bitorsors that give rise to generalised heaps are, in fact, the exact arbiters of Morita equivalence between inverse semigroups. Thus generalised heaps are of much more than merely historical interest.

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previous chapter Theory of Generalised Heaps and Generalised Groups
Footnotes
1
It is worth noting that there is a parallel here with the way that groupoids, specifically ordered groupoids, may be used to study inverse semigroups [26, Chapter 4].
 
2
See the comments on p. 25.
 
3
Cf. the definition given previously in Section 3.​6, and also Wagner’s treatment in Section 8.​2.
 
4
But we define vectors via equivalence classes of directed line segments whereas Roe [44] defines them as functions: the translations that they determine.
 
5
Cf. the definition given previously in Section 3.​6, and also Wagner’s treatment in Section 8.​3.
 
6
Johnstone [16] suggested the name ‘herdoid’ but presumably as a joke.
 
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Metadata
Title
Generalised Heaps as Affine Structures
Authors
Christopher D. Hollings
Mark V. Lawson
Copyright Year
2017
Book
Wagner’s Theory of Generalised Heaps

Print ISBN: 978-3-319-63620-7

Electronic ISBN: 978-3-319-63621-4

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-63621-4_9

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