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Walker's Cancellation Theorem

3 pagesPublished: July 28, 2014

Abstract

Walker's cancellation theorem says that if B + Z is
isomorphic to C + Z in the category of abelian
groups, then B is isomorphic to C. We construct an example in
a diagram category of abelian groups where the theorem fails. As a
consequence, the original theorem does not have a constructive
proof. In fact, in our example B and C are subgroups of
Z<sup>2</sup>. Both of these results contrast with a group whose
endomorphism ring has stable range one, which allows a
constructive proof of cancellation and also a proof in any diagram
category.

Keyphrases: Abelian groups, Constructivism, diagram category, Kripke model

In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 145--147

Links:
BibTeX entry
@inproceedings{TACL2013:Walkers_Cancellation_Theorem,
  author    = {Robert Lubarsky and Fred Richman},
  title     = {Walker's Cancellation Theorem},
  booktitle = {TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic},
  editor    = {Nikolaos Galatos and Alexander Kurz and Constantine Tsinakis},
  series    = {EPiC Series in Computing},
  volume    = {25},
  pages     = {145--147},
  year      = {2014},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {https://easychair.org/publications/paper/mk2c},
  doi       = {10.29007/vz4n}}
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