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UNDECIDABILITY AND THE DEVELOPABILITY OF PERMUTOIDS AND RIGID PSEUDOGROUPS

Part of: Special aspects of infinite or finite groups Semigroups Graph theory Algebraic structures

Published online by Cambridge University Press:  20 March 2017

MARTIN R. BRIDSON  and
HENRY WILTON
MARTIN R. BRIDSON
Affiliation:
Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK; [email protected]
HENRY WILTON
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK; [email protected]

Abstract

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A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron’s conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups.

MSC classification

Primary: 20F10: Word problems, other decision problems, connections with logic and automata 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 20M18: Inverse semigroups 08A50: Word problems
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e10
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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