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Polygraphs of finite derivation type

Published online by Cambridge University Press:  09 September 2016

YVES GUIRAUD  and
PHILIPPE MALBOS
YVES GUIRAUD
Affiliation:
INRIA Paris, IRIF, CNRS UMR 8243, Université Paris 7, Case 7014, 75205 Paris Cedex 13, France E-mail: yves.guiraud@pps.univ-paris-diderot.fr
PHILIPPE MALBOS
Affiliation:
Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France E-mail: malbos@math.univ-lyon1.fr
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Abstract

Craig Squier proved that, if a monoid can be presented by a finite convergent string rewriting system, then it satisfies the homological finiteness condition left-FP3. Using this result, he constructed finitely presentable monoids with a decidable word problem, but that cannot be presented by finite convergent rewriting systems. Later, he introduced the condition of finite derivation type, which is a homotopical finiteness property on the presentation complex associated to a monoid presentation. He showed that this condition is an invariant of finite presentations and he gave a constructive way to prove this finiteness property based on the computation of the critical branchings: Being of finite derivation type is a necessary condition for a finitely presented monoid to admit a finite convergent presentation. This survey presents Squier's results in the contemporary language of polygraphs and higher dimensional categories, with new proofs and relations between them.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 2 , February 2018 , pp. 155 - 201
Copyright
Copyright © Cambridge University Press 2016 

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