# 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

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topParkes, 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

top- R.V. Book and F. Otto, String-Rewriting Systems. Texts and Monographs in Computer Science, Springer-Verlag (1993).
- 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
- R.H. Haring-Smith, Groups and simple languages. Trans. Amer. Math. Soc.279 (1983) 337–356.
- 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
- R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer-Verlag (1977). Zbl0368.20023
- 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
- W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, 2nd edn. Dover (1976).

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