Monoid presentations of groups by finite special string-rewriting systems

Duncan W. Parkes; V. Yu. Shavrukov; Richard M. Thomas

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 38, Issue: 3, page 245-256
  • ISSN: 0988-3754

Abstract

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We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

How to cite

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Parkes, Duncan W., Shavrukov, V. Yu., and Thomas, Richard M.. "Monoid presentations of groups by finite special string-rewriting systems." RAIRO - Theoretical Informatics and Applications 38.3 (2010): 245-256. <http://eudml.org/doc/92741>.

@article{Parkes2010,
abstract = { We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group. },
author = {Parkes, Duncan W., Shavrukov, V. Yu., Thomas, Richard M.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Group; monoid presentation; Cayley graph; special string-rewriting system; word problem.; plain groups; monoid presentations; Cayley graphs; special string-rewriting systems; word problem; free products; direct products},
language = {eng},
month = {3},
number = {3},
pages = {245-256},
publisher = {EDP Sciences},
title = {Monoid presentations of groups by finite special string-rewriting systems},
url = {http://eudml.org/doc/92741},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Parkes, Duncan W.
AU - Shavrukov, V. Yu.
AU - Thomas, Richard M.
TI - Monoid presentations of groups by finite special string-rewriting systems
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 3
SP - 245
EP - 256
AB - We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.
LA - eng
KW - Group; monoid presentation; Cayley graph; special string-rewriting system; word problem.; plain groups; monoid presentations; Cayley graphs; special string-rewriting systems; word problem; free products; direct products
UR - http://eudml.org/doc/92741
ER -

References

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  1. R.V. Book and F. Otto, String-Rewriting Systems. Texts and Monographs in Computer Science, Springer-Verlag (1993).  
  2. Y. Cochet, Church-Rosser congruences on free semigroups, in Algebraic Theory of Semigroups, edited by G. Pollák. Colloquia Mathematica Societatis János Bolyai 20, North-Holland Publishing Co. (1979) 51–60.  Zbl0408.20054
  3. R.H. Haring-Smith, Groups and simple languages. Trans. Amer. Math. Soc.279 (1983) 337–356.  
  4. T. Herbst and R.M. Thomas, Group presentations, formal languages and characterizations of one-counter groups. Theoret. Comput. Sci.112 (1993) 187–213.  Zbl0783.68066
  5. R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer-Verlag (1977).  Zbl0368.20023
  6. K. Madlener and F. Otto, About the descriptive power of certain classes of finite string-rewriting systems. Theoret. Comput. Sci.67 (1989) 143–172.  Zbl0697.20017
  7. W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, 2nd edn. Dover (1976).  

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