Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-07T06:28:03.008Z Has data issue: false hasContentIssue false
  • English
  • Français

The descending chain condition on solution sets for systems of equations in groups

Published online by Cambridge University Press:  20 January 2009

M. H. Albert  and
J. Lawrence
M. H. Albert
Affiliation:
Department of Pure Mathematics, University of Waterloo, Canada
J. Lawrence
Affiliation:
Department of Pure Mathematics, University of Waterloo, Canada
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 1 , February 1986 , pp. 69 - 73
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Baumslag, B., Residually free groups, Proc. London Math. Soc. (3) 17 (1967), 402–418.CrossRefGoogle Scholar
2.Baumslag, G., Lecture Notes on Nilpotent Groups (CBMS Series Number 2, Amer. Math. Soc. 1969).Google Scholar
3.Baumslag, G. and Solitar, D., Some two generator one relator non-Hopfian groups, Bull.Amer. Math. Soc. 68 (1962), 199–201.CrossRefGoogle Scholar
4.Culik, K. II and Karhumaki, J., Systems of equations over a free monoid and Ehrenfeucht's Conjecture, Discrete Math. 43 (1983), 139–153.CrossRefGoogle Scholar
5.Culik, K. II and Salomaa, A., Test sets and checking words for homomorphism equivalence, J. of Computers and System Sciences 20 (1980), 379–395.CrossRefGoogle Scholar
6.Gersten, S. M., On fixed points of automorphisms of finitely generated free groups, Bull. Amer. Math. Soc. 8 (1983), 451–454.CrossRefGoogle Scholar
7.Lyndon, R., Equations in groups, Bol. Soc. Brasil Mat. 11 (1980), 79–102.CrossRefGoogle Scholar
8.Lyndon, R. and Schupp, P., Combinatorial Group Theory (Springer-Verlag, 1977).Google Scholar
9.Kaplansky, I., Infinite Abelian groups (U. of Michigan Press, Ann Arbor, 1969).Google Scholar
10.Makanin, G. S., Systems of equations in free groups, Siberian Math. J. 13 (1972), 402–408.CrossRefGoogle Scholar